INTERIOR POINT METHODS FOR MATHEMATICAL PROGRAMMING : A BIBLIOGRAPHY

Eberhard Kranich^{1}

Diskussionsbeitrag Nr. 171 1991

Diskussionsbeiträge des FB WiWi der FernUniversität Hagen Herausgegeben vom Dekan des Fachbereichs

1FB Wirtschaftswissenschaften und Operations Research, FernUniversität Hagen, P.O.

Box 940, D–5800 Hagen, West–Germany

Abstract

Nothing is older than a yesterday’s newspaper! In this sense, this bibliography is incomplete, of course, because of the enormous interest and ongoing research in interior–point methods for mathematical programming. These activities were initiated by Karmarkar’s polynomial–time linear programming algorithm in 1984, resulting in a flood of research papers, articles, talks, developments of commercial software and so on. This working paper should be regarded as a first trial to obtain a survey over the research reports, journal articles, etc., published so far, resp. known to the author.

Acknowledgements :

Key Words : Karmarkar’s algorithm, interior–point methods, projective scaling algorithm, affine scaling algorithm, path–following methods, logarithmic barrier function algorithm, potential reduction algorithm, trajectory– following method, polynomial–time algorithm, method of centers, gravitational method, box method, mathematical programming.

Remark : The reader is kindly requested to send those references to the authors which are not listed or incorrectly cited in this bibliography (if possible, a copy of the corresponding report or the article’s citation with the (complete!) name of the journal, with a short abstract, cross–references and keywords), or to send a copy of his own reference list (can be handwritten!) in order to complete the current bibliography^{2}. The actual list will be published elsewhere from time to time, possibly by e–mail.

Last edit : May 10, 2024

2e–mail : WI805@DHAFEU11 or WI811@DHAFEU11 (EARN), FAX : (0)2331 804313

Introduction

Interior–point methods for mathematical programming are known for a long time. With respect to solving nonlinear programming problems they have turned out to be a reliable tool, see e.g. Fiacco and McCormick[196], Frisch[227, 228],Huard[358], and Lootsma[480]. But in view of solving linear programming problems they could not compete with G. B. Dantzig’s simplex method. They were only regarded as theoretical alternative solution methods to, never as practical substitutes for the simplex method, see e.g. Brown and Koopmans[109], Caroll[111], Dikin[160, 161], Frisch[230], and Parisot[654]. That these methods have failed in linear programming practice at that time can be justified by the lack of modern numerical linear algebra^{3} and sparse matrix technologies^{4}, see e. g. Marsten et al.[511].

A new polynomial–time linear programming interior point–method was presented by N. K. Karmarkar in 1984, which was claimed to be up to 100 times faster than the simplex method, in practice and for any instance. Karmarkar’s announcement stirred the interest not only of the popular press, see e.g. Angier[21], Emmett[182], Fricker[226], Garfunkel et al.[234], Gleick[259], Kolata[448], Philpott[656], and Stipp[774], but also of the operations research community, see e.g. the surveys of Bazaraa et al.[80], pp. 416–418, Freund[220], Gill et al.[251], Goldfarb and Todd[281], Megiddo[524, 525], Monma[590], Todd[808, 812], Zimmermann[990], and, in some sense, den Hertog[333].

Since Karmarkar’s publications[403, 404], a great amount of contributions has been made towards the theoretical analysis and practical implementations of interior–point methods, see e.g. Anstreicher[30], Bayer and Lagarias[77, 78], Lagarias[466], and Monma[591], and the investigations are still going on, see the David II–Report[154], and Resende and Wright[692]. The importance of interior–points methods is reflected by numerous talks, workshops and minisymposia held at various operations research or optimization meetings, see e.g. the programs of the annual meetings of ORSA/TIMS or SIAM, Dolecki[170], Freund[220], Megiddo[526, 529], and Tovey[848, 849].

3A. George and J. W. H. Liu, Computer Solutions of Large Sparse Positive Definite Systems, Prentice Hall, 1981.

4S. Pissanetzky, Sparse Matrix Technology, Academic Press, 1984.

In order to get an overview, interior–point methods can be broadly classified into the following categories, see e.g. den Hertog[333], whereby the selected citations are thought as examples :

1. Projective scaling methods, introduced by Karmarkar[403, 404] and further investigated and extended e.g. by Akguel[17], Anstreicher[24, 25, 32, 35, 40], Asic et al.[53], Bazaraa et al.[80], Blair[93], Dennis et al.[155], Dodani and Babu[169], Ferris and Philpott[192], Franklin[206], Gay[237], de Ghellinck and Vial[244], Goldfarb and Mehrotra[278, 279], Gonzaga[292, 306], Hettich and Margraff[345], Hooker[350], Kojima[430], Kojima and Tone[447], McDiarmid[517], Mitchell[563], Mitchell and Todd[570, 571], Paris[653], Powell[677], Roos[697], Shanno[737], Shanno and Marsten[746], Shub[755], Steger[773], Todd[810], Todd and Burell[821], Tomlin[835], Vial[895], Ye[951, 956], Ye and Kojima[965], Zimmermann[992].

2. ’Pure’ affine scaling methods, originally proposed by Dikin[160, 161, 162] about twenty years ago, independently rediscovered by e.g. Barnes[63], Cavalier and Soyster[116], Monma and Morton[592], Vanderbei et al.[888], and further investigated by e.g. Adler et al.[9], Barnes et al.[72] (belongs in principle to the class of path–following methods), Cavalier and Schall[115], Christiansen and Kortanek[138, 139], Hettich and Ries[346], Kortanek and Shi[457], Marsten et al.[507], Mehrotra[540], Monteiro et al.[601] (the polynomiality is derived from the polynomial result of path–following methods !), Ponnambalam and Vanelli[671], Resende and Veiga[690], Sherali[751] (in principle a path–following method), Strang[781, 782, 783], Todd[819], Wei[900, 901]. Megiddo and Shub[532] have shown that algorithms belonging to this class can be supposed to be non–polynomial.

3. Path– or trajectory–following methods, initiated by Gill et al.[253], showing a certain equivalence to Karmarkar’s algorithm, and further investigated by e.g. Anstreicher et al.[43], Ben–Daya and Shetty[83, 82], Goldfarb and Liu[275], Gonzaga[291, 302], Gonzaga and Todd[308], den Hertog et al.[335], Kojima et al.[443], Lustig[484], Lustig et al.[494, 492], Marsten et al.[511], Marxen[513], McShane et al.[520], Megiddo[528], Mehrotra[542], Mehrotra and Sun[549], Monteiro and Adler[598, 599], Osborne[636, 637], Renegar[682], Roos[704, 705],Todd[817], Todd and Vial[824], Tseng[853], Vaidya[869], Zhu and Kortanek[986].

4. Affine scaling methods applied to a potential function, described in Anstreicher and Bosch[41], Freund[221], Gonzaga[295, 299, 303], Ye[959].

5. Method of centers, described e.g. in Boggs et al.[100, 102], Jarre[375, 378, 381], Jarre et al.[383], Mehrotra and Sun[552], Sonnevend[763, 765, 766], Witzgall et al.[908].

6. Gravitational method, developed in Murty[612, 613] and implemented by Chang and Murty[121].

7. Box method, as described in Zikan and Cottle[987, 988, 989].

Algorithms belonging to the class 4 are the most promising, because they do not loose their theoretical complexity in practice. Unfortunately, no computational experiences are published, at least as yet.

During the last few years practical implementations of linear programming interior–point methods were developed, which are superior to the simplex method, not necessarily for each special problem, but at least for a set of problems, such as Gay’s netlib problem collection, see e.g. Lustig[486], Marsten et al.[507, 511], McShane et al.[520]. For solving everyday linear programming problems there exists a rule of thumb : If the problem size is small, i.e. the number of variables is lower than 500, the simplex method is superior in most cases and consequently ’still alive’, otherwise solve the problem via an interior– point method. Nowadays, there exist at least two commercial implementations. The first one is OB1, developed by Shanno and Marsten[747], see also e.g. Lustig et al.[497], and the second is the AT & T KORBX Linear Programming System[55, 126], a software/hardware combination.

A concluding remark : Talks, workshops and minisymposia are handled as technical reports, because I am rather sure that to each talk there exists an underlying report.

Title word cross-reference

Big − M [445]. L [670]. L_{1} [714, 752]. L_{∞} [714]. log x [117]. O((m + n)n^{2} + (m + n)^{1.5}nL) [869]. O(n^{0.5}L) [342]. O(n^{3.5}L) [549]. O(n^{3}L) [580, 583, 582, 955, 41, 275, 291, 543, 578, 596, 705, 959]. O(n^{ρ}L) [580]. O(nL) [843, 308, 298, 446, 519, 822, 963, 969]. P [440, 938]. P_{0} [652].

1990s [154].

48 [138].

5th [170].

’91 [616].

absolute [555]. Accelerating [797, 799]. Acceleration [627]. acceptance [25]. accuracy [500, 832]. Active [898, 845]. adaptation [429, 890]. adaptive [382, 426, 582, 588]. Adding [240]. Adopting [18]. advanced [284]. Advances [37, 190, 283, 449, 497]. Affine [73, 264, 263, 670, 877, 12, 14, 45, 54, 58, 71, 72, 74, 68, 63, 70, 91, 115, 138, 139, 176, 188, 207, 246, 308, 304, 294, 307, 299, 327, 372, 373, 371, 452, 455, 473, 505, 506, 507, 512, 536, 537, 540, 557, 569, 592, 597, 601, 671, 672, 679, 688, 690, 687, 784, 819, 822, 855, 858, 860, 859, 875, 878, 887, 906, 952, 939, 949, 973, 975, 152, 153, 77]. against [146]. ähnliche [423]. airlift [110, 425, 904]. Algebra [242, 257, 916]. algebraic [1]. algoritam [49]. Algorithm [148, 2, 9, 13, 3, 10, 7, 5, 8, 16, 22, 29, 35, 41, 44, 36, 39, 28, 23, 42, 24, 25, 26, 33, 40, 32, 46, 47, 58, 59, 62, 72, 74, 68, 61, 63, 70, 79, 82, 83, 84, 89, 87, 93, 94, 96, 97, 103, 104, 107, 108, 114, 115, 116, 120, 118, 122, 123, 124, 132, 138, 139, 141, 145, 155, 157, 158, 163, 164, 165, 166, 175, 180, 182, 185, 191, 192, 193, 206, 207, 208, 223, 224, 214, 219, 209, 212, 222, 213, 235, 236, 237, 240, 245, 260, 261, 269, 266, 270, 268, 274, 275, 278, 277, 280, 281, 276, 296, 308, 287, 290, 304, 294, 307, 291, 313, 315, 323]. algorithm [326, 327, 329, 331, 332, 345, 346, 350, 355, 356, 357, 365, 368, 371, 374, 384, 385, 392, 388, 390, 394, 393, 397, 401, 403, 417, 422, 418, 421, 420, 406, 404, 405, 414, 415, 416, 430, 447, 439, 443, 446, 442, 451, 452, 457, 454, 455, 462, 463, 468, 469, 471, 477, 475, 476, 472, 478, 482, 489, 484, 483, 502, 505, 508, 507, 515, 518, 519, 521, 527, 533, 550, 549, 535, 543, 557, 558, 560, 561, 563, 574, 564, 566, 569, 567, 572, 570, 571, 568, 573, 576, 578, 583, 582, 579, 594, 596, 600, 597, 601, 602, 603, 623, 624, 631, 630, 634, 635, 647, 649, 646, 652, 653, 657, 658, 659, 673]. algorithm [669, 677, 679, 682, 684, 688, 690, 687, 689, 695, 700, 702, 704, 714, 718, 719, 722, 725, 726, 729, 733, 736, 734, 735, 745, 746, 751, 752, 755, 757, 758, 759, 760, 762, 773, 778, 777, 781, 783, 785, 786, 802, 809, 817, 821, 822, 827, 818, 814, 824, 826, 807, 806, 815, 805, 810, 816, 829, 828, 832, 831, 830, 843, 846, 847, 848, 849, 853, 854, 855, 865, 866, 870, 869, 871, 874, 875, 878, 887, 888, 891, 892, 906, 910, 917, 943, 956, 953, 927, 965, 970, 971, 972, 938, 946, 939, 926, 957, 944, 968, 955, 948, 963, 969, 961, 931, 932, 934, 959, 973, 974, 979, 978, 986, 985]. algorithme [426, 472, 829, 828, 831]. Algorithmen [310]. Algorithmic [100, 751]. Algorithms [150, 151, 184, 651, 1, 4, 6, 18, 20, 30, 38, 31, 54, 73, 86, 98, 110, 119, 127, 143, 189, 215, 214, 221, 239, 246, 265, 295, 289, 293, 300, 299, 292, 298, 310, 311, 319, 318, 316, 325, 328, 343, 353, 354, 387, 399, 413, 428, 431, 434, 445, 438, 437, 450, 458, 465, 479, 481, 495, 512, 518, 532, 541, 580, 584, 581, 585, 587, 586, 589, 588, 591, 598, 599, 621, 626, 630, 644, 645, 648, 670, 674, 706, 720, 754, 763, 765, 779, 801, 808, 823, 863, 895, 936, 942, 952, 954, 945, 949, 958, 940, 966, 960, 933, 980, 981, 977]. algorithmu [374]. Algorithmus [260, 725]. algoritmo [148]. allocation [409, 876]. allowing [948]. along [699, 703]. ALPO [880, 879]. Amelioration [828]. Analiza [49]. analogous [208]. Analyse [502]. Analysis [22, 23, 502, 96, 139, 483, 486, 512, 595, 683, 751, 752, 795, 861, 912, 918, 952, 953, 960, 212]. Analytic [465, 710, 268, 381, 376, 378, 383, 375, 377, 552, 763, 765, 767, 769, 766, 768, 896, 941]. angular [90, 129, 731]. Ansätze [156]. Anstreicher [810]. Anticipated [584, 585, 587, 586, 809, 813, 823, 949]. apparatus [76, 409, 876]. Application [141, 422, 767, 57, 112, 232, 286, 454, 654, 673, 752, 759]. Applications [766, 768, 84, 90, 110, 129, 359, 425, 481, 764, 790]. Applied [782, 13, 190, 381, 783]. appraisal [168]. approach [173, 187, 201, 202, 246, 311, 320, 360, 410, 419, 408, 438, 437, 482, 608, 680, 686, 685, 835, 895, 907, 931]. Approaches [523, 254, 324, 407, 473, 522, 964]. approchee [654, 787]. Approximate [474, 654, 657, 894, 25, 57, 659, 709, 711, 708, 705, 787, 536]. approximately [530]. approximating [427]. approximation [606, 766, 957]. approximations [769]. architecture [125]. architectures [495]. Arising [467, 248, 638, 768]. arithmetic [549, 869]. art [740, 739]. Asilomar [692]. Asimptotsko [50]. aspect [323]. aspects [58, 247, 415, 647]. assignment [46, 973]. associated [176, 366, 795]. assumption [752]. assumptions [38, 952]. Asymptotic [53, 51, 679, 96, 755, 756]. attack [840]. attempt [897]. attend [692]. augmented [201, 202]. available [486].

backpropagation [851]. Band [81]. Barrier [194, 666, 667, 925, 43, 27, 82, 83, 163, 222, 252, 255, 256, 247, 253, 250, 276, 343, 335, 338, 369, 389, 452, 455, 459, 513, 514, 610, 636, 641, 664, 661, 660, 712, 706, 743, 843, 898, 907, 912, 302]. barriers [565, 562]. based [163, 207, 219, 221, 209, 260, 311, 343, 360, 550, 640, 682, 706, 765, 896, 952]. bases [531]. BASIC [693, 134, 430, 956, 211, 987]. Basics [242]. basierend [260]. basis [429, 470, 489, 485, 672, 798, 801]. be [632]. behaved [868]. Behavior [52, 328, 12, 14, 53, 51, 54, 318, 330, 393, 441, 532, 584, 585, 587, 586, 593, 679, 755, 756, 809, 813, 823, 908, 949, 966]. Beispiel [725]. Bell [55]. Berkeley [463]. better [406, 145]. between [432, 569]. Beyond [496]. bibliography [464]. Bimatrix [935]. bisection [388]. block [90, 129, 731, 846, 847]. bornees [828]. bound [103, 297]. Boundary [532, 756, 180, 593, 966]. bounded [191, 505, 773, 828, 988]. bounding [359]. bounds [564, 775, 807]. Bowdoin [467]. box [323, 647, 649, 694, 987, 988, 989]. branch [103]. Breakthrough [259]. breakthroughs [590]. brief [879, 911]. Brunswick [467]. build [152, 153, 309, 337, 936, 950, 930, 964]. build-down [936, 950, 930]. buys calculation [839]. call [511]. can [632]. Canonical [391, 856]. Carlo [6]. cas [828]. case [32, 92, 146, 329]. Castel [170]. Center [791, 102, 100, 216, 223, 224, 225, 214, 210, 212, 553, 660, 708, 705, 763, 766, 790, 867, 868, 909]. Centered [793, 794, 792, 791, 827]. centering [71, 65, 73, 70, 208, 209]. centers [66, 172, 183, 268, 339, 358, 359, 381, 376, 378, 383, 375, 377, 429, 432, 552, 709, 711, 710, 767, 768, 896, 941]. Central [453, 224, 267, 384, 558, 770, 771]. centre [765]. centres [905]. certain [872]. challenge [251, 849]. Characteristics [789]. Cholesky [238, 284, 387, 537, 846, 847, 891]. class [13, 20, 119, 164, 208, 313, 336, 335, 396, 442, 600, 707, 846, 847, 921, 923, 937, 951, 967, 981, 977, 985, 992]. classes [632, 645, 763]. classical [145, 335, 604]. coarse [18]. code [497, 738]. col [148]. collapse [138, 139]. College [467]. column [269, 387, 565, 562, 948]. columns [285, 882, 883, 884, 942]. Combinatorial [398, 622, 563, 570, 571, 573, 941]. combined [29, 28, 219, 825, 810]. Combining [814, 92]. command [904]. comments [405]. Committee [151, 150]. Comparative [952, 426, 471]. compared [789]. Comparing [98, 176, 836]. comparison [45, 473]. comparisons [183]. Complementarity [792, 163, 164, 314, 316, 315, 386, 431, 433, 440, 438, 441, 435, 436, 439, 444, 446, 442, 437, 578, 583, 582, 589, 577, 614, 613, 632, 631, 630, 646, 652, 676, 818, 967, 938, 957, 961]. complementary [319]. complete [238, 410, 408, 537]. complex [226]. Complexity [130, 775, 953, 36, 26, 341, 342, 384, 386, 427, 525, 535, 683, 770, 771, 776, 817, 824, 853, 872, 921, 982, 983]. composite [803]. Computation [139, 133, 268, 271, 540]. Computational [3, 5, 109, 108, 123, 136, 183, 323, 349, 360, 397, 416, 494, 592, 647, 689, 750, 45, 46, 116, 167, 168, 169, 352, 354, 415, 472, 492, 510, 659, 741, 740, 840, 571]. computationally [420, 831]. computations [643]. compute [239]. computer [19]. computers [18, 238, 503]. Computing [66, 90, 129, 138, 737, 866, 57, 284, 415, 553, 650, 897, 957]. concepts [156]. concerning [68]. Conference [170, 467, 526]. Conical [292, 288, 287, 289]. conjugate [128, 536, 608, 735, 891]. considerations [352]. constrained [114, 200, 208, 209, 276, 301, 307, 332, 552, 553, 627, 645, 743, 784, 913, 917, 915, 923, 933, 986]. constraint [285, 761, 910]. constraints [85, 104, 117, 207, 300, 323, 358, 647, 649, 694, 906, 914]. contain [444]. Containing [970, 69, 807]. Contemporary [234]. Continuation [674, 433, 440, 441, 435, 436]. continuous [12, 14, 594]. control [417, 768]. controlled [453]. controllers [767]. Controlling [459]. Convergence [206, 255, 294, 318, 375, 393, 457, 593, 74, 68, 70, 91, 122, 161, 304, 381, 386, 434, 461, 474, 483, 544, 600, 636, 639, 751, 752, 754, 799, 854, 855, 858, 864, 859, 862, 863, 887, 908, 958, 968, 980, 981, 976, 977]. convergent [214, 344, 519, 631, 630, 800, 802, 922, 963, 969, 979, 978]. convex [13, 43, 83, 113, 187, 207, 228, 274, 275, 276, 307, 336, 335, 339, 381, 376, 378, 379, 375, 377, 380, 384, 390, 399, 400, 456, 459, 479, 552, 550, 549, 551, 600, 601, 627, 680, 707, 760, 764, 763, 765, 769, 768, 784, 826, 852, 870, 871, 971, 972, 933, 932, 986, 985, 599]. convexity [362]. coordinates [78, 466]. core [238]. Cornell [144]. corner [21]. correcting [280, 533]. correction [172]. corrector [112, 164, 490, 545, 541, 803, 824]. cost [231, 424, 515, 687, 689]. costs [608]. covering [66, 420]. Cray [138, 499, 491, 506]. created [111]. crew [75]. criteria [620]. criterion [25, 832]. cubically [800]. Current [738, 796, 927, 925]. curriculum [848]. curvature [982]. Cutting [269, 927, 926, 749, 928].

d [174]. Dampf [772]. dans [426, 828]. Dantzig [815]. Data [8]. decision [243]. Decomposed [846, 847]. Decomposition [265, 266, 896, 267, 394, 428]. defense [190]. Deferred [543]. definite [444, 885]. Degeneracy [317, 557, 33, 805, 816, 875]. Degenerate [860, 52, 858, 859, 861, 862]. Dense [882, 285, 419, 883, 884]. dependencies [229]. Derivation [389, 472]. derive [472]. descent [95, 121, 429, 723, 818]. description [879]. design [286, 767]. details [741]. Detecting [134]. Determining [430]. development [137, 617, 932]. Developments [467, 31, 198, 421, 407, 414, 534, 780, 812, 878, 902, 944, 990]. diagonal [606]. different [639]. difficult [420]. Dikin [149, 723]. Dikin’s [633, 887]. dimensional [171, 474]. direction [615, 761, 819]. directions [44, 306, 334, 487, 569, 819, 812, 991, 992]. directly [215, 213]. Discrete [655]. Discussion [984]. displacement [138]. distance [665]. double [227]. down [936, 950, 930]. Dual [33, 636, 856, 931, 2, 45, 58, 84, 113, 136, 137, 185, 252, 249, 261, 268, 308, 355, 356, 357, 382, 434, 443, 452, 455, 453, 499, 491, 493, 487, 484, 494, 492, 500, 505, 510, 506, 507, 520, 519, 527, 531, 542, 544, 541, 588, 592, 596, 598, 599, 597, 601, 626, 671, 672, 688, 690, 687, 733, 734, 735, 750, 738, 760, 788, 802, 821, 822, 923, 965, 966, 973, 980, 978]. duale [261]. duales [500]. dynamic [924]. dynamical [795].

early [774]. easy [98]. editor [162]. Effects [719, 805, 816]. efficaces [831]. efficiency [56, 222, 218]. Efficient [1, 642, 850, 883, 124, 409, 447, 456, 476, 603, 643, 831, 876, 892]. Effizienz [56]. Elaboration [787, 831, 787]. electronic [59]. Eliminating [942]. ellipsoid [406, 427, 633, 718, 769, 931, 934]. Ellipsoids [444, 33, 69, 66, 268, 364, 807, 970]. empirical [110]. enables [166]. endlicher [500]. enforced [792]. engineering [326]. English [787]. enhancements [100]. entropy [117, 184, 186, 325, 968]. equality [200]. equations [16, 87, 245, 277, 536]. equilibrium [935]. equivalence [253]. Erfahrungen [725]. Estimating [982]. estimation [752, 766]. Etude [426]. evaluation [110]. evening [785]. exact [16, 787, 900]. exacte [787]. example [195, 725]. Existence [314, 140]. experience [116, 123, 136, 349, 360, 397, 494, 492, 510, 592, 659, 750, 741]. Experiences [726, 148, 725]. experienze [148]. experiment [46]. Experimental [575, 834, 483, 835]. Experimentation [691]. Experiments [633, 20, 345, 451, 457, 560, 470]. expert [141]. explicit [122]. explicitly [988]. Exploiting [120, 806]. extension [209, 245, 400, 600, 601, 641, 773, 821, 943, 972]. Extensions [361]. exterior [794, 899]. extrapolation [765, 770, 771]. extreme [140].

face [603]. faces [1, 850]. facet [554]. facilities [422]. factor [537]. factorization [284, 387, 846, 847]. factorizations [238]. family [44]. Fast [399, 643, 365, 368, 448, 516, 798, 873]. faster [226, 537]. Feasibility [488, 388, 894]. feasible [470, 615, 761]. feedback [767]. few [104]. Fiacco [511]. fields [138, 176, 366]. fill [87]. fill-in [87]. finding [210, 212, 531, 554, 548, 603, 868]. Finite [544, 500, 958]. First [713, 558]. flattening [791]. flow [27, 129, 135, 231, 424, 515, 608, 688, 687, 689, 993]. Flows [80, 95, 399, 401, 559, 690]. focuses [258]. Folding [21]. following [4, 34, 296, 305, 293, 302, 303, 316, 341, 342, 340, 337, 384, 450, 478, 551, 558, 583, 582, 585, 598, 599, 627, 681, 700, 702, 704, 770, 771, 824, 823, 854, 914, 920, 936, 942, 970, 975, 982, 986, 985]. force [311]. form [35, 36, 26, 152, 153, 227, 229, 237, 385, 684, 856, 988]. formulation [260, 498]. Formulierung [260]. FORTRAN [286]. Fractional [918, 23, 24]. framework [433, 575, 745]. France [170]. Free [243, 546, 752, 877]. French [170]. Freund [857]. Fully [893, 957]. function [82, 83, 109, 159, 177, 186, 195, 221, 222, 213, 274, 276, 302, 343, 335, 361, 362, 364, 368, 369, 370, 474, 479, 610, 641, 665, 664, 712, 698, 702, 706, 843, 868, 870, 871, 907]. functions [163, 207, 359, 389, 390, 440, 636, 661, 898, 918, 937, 947, 935]. Further [137, 421, 414, 944, 964, 932, 938].

GAUSS [84]. Genauigkeit [500]. general [91, 216, 320, 433, 468, 790, 981]. Generalization [390, 209, 261, 778]. generalization/extension [209]. Generalized [545, 375, 384, 389, 608, 684, 957]. generate [319]. generation [269, 565, 562, 635, 948]. generic [484]. geometric [11, 452, 455, 453]. geometry [77, 78, 219, 411, 466]. German [170]. get [186]. Ghellinck [261]. Global [769, 795, 854, 858, 859, 190, 600, 631, 640, 763, 863, 889, 890, 954]. globally [630, 968]. Gordan [389]. gradient [128, 227, 395, 536, 608, 611, 629, 735, 746, 891]. gradients [221]. grain [18]. grande [472]. graph [686]. graphs [419]. gravitational [121, 612]. gravity [183]. group [929].

Hamiltonian [95]. Handling [546, 883]. hard [448]. held [144, 467, 713]. Higher [99, 112, 547, 539]. Hilbert [466]. historical [617]. history [911].

Holmdel [55]. homogeneous [16, 245, 277, 390]. homotopies [618]. Homotopy [435, 436, 619, 97, 232, 379, 621, 715, 769]. Huard [905]. hybrid [457, 454, 576, 841].

I. [887]. ICIAM [616]. idea [725]. Idee [725]. identification [764, 798, 801]. Identifying [485, 181]. ihre [776]. II [212, 599, 29, 28, 78, 219, 303, 571, 825, 814, 810, 838, 988]. III [466]. im [260]. Imai [367]. Imai’s [861, 862, 976]. IMP [503]. Implementation [4, 7, 58, 148, 415, 507, 534, 536, 537, 558, 560, 602, 675, 745, 760, 833, 2, 9, 8, 59, 124, 128, 247, 284, 309, 331, 383, 447, 471, 489, 506, 520, 542, 535, 538, 541, 561, 595, 688, 720, 726, 741, 834, 842, 891, 906]. Implementations [540, 60, 537, 591, 691]. Implementazione [148]. implemented [127, 785]. Implementierung [331]. Implementing [10, 114, 115, 658, 722, 837, 838, 15, 490, 508, 744]. IMPLO [286]. Improved [807, 36, 392]. improvement [517]. improves [474]. Improving [461, 828]. indefinite [248, 886]. independent [419]. indicators [181]. inductive [685]. inequalities [106, 109, 327, 764, 765]. inequality [114, 117, 208, 209, 300, 854]. inexpensive [672]. infeasible [215, 217, 213, 218]. inference [685]. infinite [48, 193, 454, 678]. influence [702]. Inner [460, 718, 724]. innere [500, 776]. inscribed [364, 427]. insights [751]. Integer [398, 622, 728, 103, 418, 421, 684, 907]. Integrability [366]. integrals [982]. Integrating [848]. Integration [470]. Intelligent [629]. interaction [495]. interactive [166]. interest [757]. intérieures [787]. Interior [133, 142, 215, 220, 241, 251, 271, 274, 300, 312, 319, 324, 333, 380, 412, 424, 464, 479, 496, 511, 504, 509, 529, 598, 599, 644, 648, 663, 676, 713, 716, 736, 734, 735, 740, 739, 747, 852, 851, 899, 913, 915, 954, 945, 933, 13, 15, 10, 7, 6, 18, 19, 30, 38, 37, 48, 60, 65, 64, 67, 75, 84, 87, 88, 92, 101, 103, 113, 112, 134, 136, 137, 135, 143, 152, 153, 149, 174, 173, 175, 177, 181, 186, 193, 199, 201, 202, 214, 211, 212, 231, 239, 243, 258, 265, 282, 273, 275, 285, 284, 286, 306, 301, 309, 313, 311, 314, 318, 315, 317, 320, 323, 326, 332, 334, 366, 386, 387, 397, 400, 418, 421, 420]. interior [410, 419, 408, 411, 416, 431, 434, 445, 438, 439, 443, 437, 456, 461, 468, 470, 473, 499, 497, 491, 490, 493, 495, 487, 485, 484, 483, 488, 494, 492, 498, 500, 501, 505, 510, 506, 507, 515, 520, 532, 527, 542, 550, 546, 543, 534, 538, 548, 559, 560, 565, 562, 575, 579, 581, 586, 588, 591, 596, 595, 602, 605, 607, 608, 611, 617, 627, 632, 630, 637, 640, 647, 649, 665, 668, 673, 669, 675, 684, 692, 686, 691, 689, 717, 720, 722, 721, 730, 731, 733, 749, 741, 738, 742, 748, 753, 754, 756, 761, 775, 788, 797, 798, 799, 802, 801, 803, 813, 817, 820, 815, 833, 836, 837, 838, 845, 863, 886, 882, 889]. interior [890, 900, 901, 917, 916, 919, 920, 921, 924, 958, 968, 940, 960, 932, 980, 981, 979, 977, 978, 993, 1, 154, 850, 776]. International [713]. Introduction [147, 234, 347, 351, 522, 782]. intuitive [219]. Inverse [338]. investigation [575, 689]. Iri [367, 861, 862, 976]. Issues [912, 4, 130, 488, 557, 916]. iteration [19, 308, 298, 446, 517, 519, 580, 822, 843, 963, 969]. iterations [40]. Iterative [160, 93, 161, 185, 415, 631, 630, 717]. iteratively [779]. Ithaca [144].

January [713]. Joint [467]. July [467]. June [467]. June/July [467]. Just [511].

Kalman [188]. Karmarkar [49, 79, 145, 157, 167, 168, 169, 182, 260, 281, 331, 350, 374, 423, 392, 472, 502, 563, 604, 618, 677, 725, 778, 777, 781, 783, 829, 828, 831, 934, 2, 9, 3, 5, 8, 17, 22, 35, 44, 23, 25, 40, 32, 46, 47, 52, 53, 51, 59, 62, 61, 89, 90, 93, 94, 96, 107, 116, 120, 118, 122, 123, 124, 129, 132, 141, 155, 158, 159, 192, 206, 208, 209, 235, 236, 237, 240, 245, 253, 270, 272, 279, 278, 277, 280, 290, 328, 330, 329, 345, 360, 362, 385, 389, 393, 400, 401, 417, 422, 413, 415, 426, 430, 447, 462, 463, 469, 471, 477, 474, 475, 476, 472, 482, 508, 517, 516, 518, 521, 533, 535]. Karmarkar [538, 561, 574, 564, 566, 569, 567, 572, 570, 571, 568, 573, 592, 600, 623, 635, 640, 653, 657, 658, 659, 671, 672, 673, 678, 679, 695, 697, 698, 714, 719, 723, 726, 727, 729, 737, 744, 745, 746, 751, 758, 759, 773, 785, 786, 789, 795, 804, 809, 821, 807, 806, 805, 816, 829, 828, 832, 831, 830, 835, 842, 844, 846, 847, 848, 856, 865, 866, 874, 888, 891, 918, 943, 956, 953, 965, 972, 930, 931, 974, 984, 815, 849]. Karmarkarova [374]. Karmarkarove [50]. Karmarkar’s [390]. KKT [250]. knapsack [651]. Kojima [42, 818]. Komplexe [226]. Komplexität [776]. Konzepte [156]. KORBX [128, 127, 125, 126, 321, 351, 352, 353, 881, 55, 110, 425].

Laboratories [55, 233]. Lagrange [511]. language [84]. Large [144, 295, 302, 303, 431, 475, 668, 75, 92, 99, 178, 200, 265, 285, 284, 308, 304, 294, 357, 416, 434, 472, 504, 558, 606, 609, 649, 666, 667, 669, 695, 822, 843, 846, 847, 897, 929]. larger [296]. Law [150, 151]. Least [555, 178, 201, 202, 643, 734, 779]. Legendre [78]. Legrende [466]. Leontief [924]. Letter [162, 233]. library [286]. like [815]. limit [139, 819]. Limiting [12, 14, 441, 54]. Line [610, 946]. lineaire [654, 787]. lineaires [472, 828]. Linear [27, 55, 80, 128, 127, 125, 148, 203, 229, 241, 242, 257, 348, 354, 402, 467, 523, 524, 614, 613, 656, 696, 713, 699, 703, 728, 844, 867, 905, 916, 9, 4, 12, 14, 11, 15, 10, 7, 16, 20, 22, 29, 41, 36, 39, 28, 34, 23, 38, 42, 31, 37, 24, 26, 33, 53, 51, 56, 62, 73, 79, 77, 78, 82, 86, 89, 88, 92, 93, 95, 97, 102, 100, 99, 101, 104, 106, 109, 108, 114, 115, 117, 119, 118, 121, 131, 133, 136, 140, 143, 145, 152, 153, 154, 155, 156, 160, 163, 164, 165, 167, 168, 169, 171, 172, 173, 175, 176, 180, 185, 186, 189, 188, 187, 190, 193, 198, 199, 200, 201, 202, 205]. linear [204, 206, 207, 208, 215, 217, 223, 220, 219, 221, 209, 211, 222, 213, 218, 230, 227, 232, 235, 236, 237, 239, 240, 243, 244, 245, 246, 252, 251, 254, 255, 256, 253, 249, 250, 261, 262, 270, 267, 282, 273, 278, 277, 285, 284, 286, 296, 295, 308, 287, 297, 304, 294, 305, 306, 293, 300, 299, 292, 291, 302, 303, 298, 309, 313, 312, 311, 317, 333, 341, 343, 334, 340, 338, 344, 349, 350, 355, 356, 361, 363, 364, 365, 368, 369, 370, 367, 372, 373, 371, 374, 382, 385, 386, 423, 388, 394, 393, 395, 396, 403, 407, 404, 405, 411, 414, 415, 416, 428, 431, 434, 438, 439, 443, 444, 446, 442, 437, 432, 449, 451, 457, 458, 460]. linear [461, 465, 466, 472, 478, 490, 487, 485, 488, 486, 494, 492, 500, 501, 506, 507, 511, 504, 509, 512, 513, 514, 517, 518, 520, 519, 528, 532, 522, 525, 530, 545, 547, 554, 534, 538, 536, 558, 560, 564, 566, 565, 569, 567, 572, 562, 576, 578, 580, 583, 584, 582, 579, 585, 587, 586, 589, 588, 577, 590, 592, 594, 596, 600, 601, 593, 606, 609, 615, 611, 612, 619, 620, 617, 621, 624, 625, 626, 629, 632, 630, 634, 635, 636, 637, 638, 639, 641, 642, 643, 646, 652, 653, 654, 665, 661, 662, 668, 673, 674, 679, 680, 682, 681, 689, 694, 695, 697, 709, 711, 710, 712, 708, 701, 700, 702, 704, 705, 706, 718]. linear [727, 729, 733, 736, 734, 740, 739, 742, 747, 751, 752, 754, 755, 756, 758, 762, 761, 764, 763, 770, 771, 773, 775, 780, 788, 794, 791, 795, 798, 800, 802, 801, 819, 813, 817, 821, 822, 825, 827, 818, 814, 808, 823, 811, 807, 806, 815, 812, 810, 816, 828, 833, 835, 836, 841, 843, 845, 853, 854, 858, 864, 860, 857, 856, 859, 861, 862, 863, 865, 869, 872, 873, 880, 877, 888, 883, 884, 892, 894, 895, 896, 898, 899, 900, 901, 902, 906, 908, 909, 910, 911, 918, 919, 920, 922, 924, 929, 937, 942, 950, 951, 956, 927, 965, 967, 938, 946, 926, 957, 949, 958, 962, 968, 955, 925, 930, 966, 963, 969, 960, 928, 964, 961, 933, 959, 980, 979, 976, 977, 975, 978, 982]. linear [983, 987, 988, 989, 990, 598]. lineare [148]. Linearen [156, 423, 56, 261, 262]. linearer [500]. linearly [301, 307, 627, 743, 784, 917, 933, 986]. linearnim [374]. linesearch [35]. little [238, 445]. Local [864, 861, 863]. locally [868, 968]. locating [800]. Log [743]. logarithmic [163, 228, 227, 276, 343, 335, 389, 452, 455, 459, 610, 654, 712, 706, 843]. logarithmique [654]. logiciel [472]. logiciels [831]. Long [41, 712, 43, 34, 341, 343, 342, 478, 583, 584, 587]. lösen [226]. Lösung [56, 500]. low [817, 824]. lower [297, 564]. LP [60, 138, 139, 149, 224, 219, 247, 319, 337, 346, 357, 430, 447, 597, 595, 683, 720, 722, 726, 846, 847, 882, 878]. LP-problem [247]. LP’s [505, 18].

machines [675]. macros [348]. made [656]. main [17]. Maine [467]. Management [147]. many [104]. Massive [238]. massively [18, 19]. matching [574, 568]. material [138]. math [774, 107]. MATH/PROTRAN [107]. Mathematical [467, 529, 685, 184, 288, 310, 320, 358, 464, 526, 633, 792, 793, 837, 838, 977]. Mathematics [234, 782, 903, 154, 783]. Mathematik [322, 903]. mathematische [310]. matrices [885]. matrix [61, 285, 652, 873, 938]. maximal [427]. maximization [628]. maximizing [109]. maximum [419]. McCormick [511]. mechanized [229]. Mehrotra [490]. memory [238].

Method [893, 19, 43, 45, 52, 53, 51, 48, 56, 57, 71, 75, 79, 100, 101, 103, 106, 113, 121, 132, 136, 137, 135, 145, 152, 153, 149, 167, 168, 169, 174, 172, 183, 186, 201, 202, 203, 216, 223, 225, 228, 230, 227, 229, 231, 244, 252, 251, 255, 247, 253, 262, 272, 282, 273, 279, 301, 302, 303, 309, 330, 336, 335, 340, 339, 337, 344, 358, 359, 364, 369, 370, 367, 372, 373, 381, 382, 376, 378, 379, 383, 375, 377, 386, 387, 400, 406, 415, 429, 440, 441, 470, 474, 499, 491, 490, 493, 487, 496, 485, 488, 494, 492, 500, 510, 506, 517, 516, 520, 542, 552, 550, 565, 562, 575, 577, 592, 604, 606, 607, 615, 611, 612, 618]. method [627, 633, 637, 638, 639, 640, 641, 671, 672, 678, 682, 694, 695, 697, 707, 709, 711, 698, 708, 705, 727, 730, 731, 750, 749, 744, 761, 764, 784, 788, 789, 790, 792, 793, 794, 791, 795, 800, 803, 824, 834, 835, 837, 838, 840, 841, 842, 845, 860, 857, 859, 861, 862, 868, 889, 890, 894, 900, 901, 905, 918, 919, 920, 922, 923, 924, 943, 927, 925, 930, 928, 931, 934, 976, 975, 984, 987, 988, 989, 993, 857]. Methodes [787]. Methodolgy [481]. Methods [76, 81, 142, 197, 241, 409, 480, 529, 609, 693, 713, 876, 15, 37, 57, 60, 65, 67, 85, 88, 92, 99, 108, 112, 131, 133, 134, 154, 156, 171, 172, 177, 181, 194, 195, 199, 200, 205, 204, 220, 211, 212, 243, 251, 256, 253, 249, 250, 258, 264, 263, 271, 285, 284, 286, 288, 297, 305, 306, 302, 303, 312, 317, 320, 333, 334, 342, 338, 366, 380, 391, 396, 412, 411, 424, 433, 435, 436, 456, 460, 461, 464, 473, 495, 498, 501, 511, 504, 509, 513, 514, 551, 545, 547, 546, 544, 539, 534, 538, 536, 537, 548, 540, 559, 590, 595, 617, 625, 632, 663, 660, 666, 667, 668, 676, 675, 681, 683, 692, 716]. methods [717, 741, 743, 740, 739, 742, 747, 748, 753, 756, 769, 776, 775, 779, 797, 798, 799, 813, 825, 833, 836, 839, 840, 852, 850, 851, 858, 864, 886, 882, 899, 912, 913, 914, 916, 915, 921, 926, 962, 982, 991, 992, 721]. metode [50]. metric [155, 865]. military [110, 425, 904]. Minimization [196, 200, 289, 474, 670, 714]. minimizing [870, 871]. Minimum [69, 66, 231, 424, 515, 687, 689]. mise [787]. mixed [907]. Mizuno [42, 818]. model [984]. modelling [904]. models [360, 417, 811]. modification [116, 157, 260, 502, 630, 888]. Modified [665, 664, 661, 660, 22, 35, 346, 641, 786, 974]. modifikacije [49]. Modifikation [260, 502]. monotone [223, 314, 316, 315, 441]. monotonic [24, 224]. monotonicity [36, 39]. Monte [6]. Motivation [156]. MP [138]. MP/48 [138]. MPS [151]. multi [360, 673]. multicommodity [129, 135, 399, 401, 559]. multiobjective [602, 852]. multiple [1, 205, 850]. multiplex [230, 229, 762]. multiplication [873]. multiplicative [361, 362, 364, 368, 369, 370]. multistage [607].

N. [93, 729, 804]. Naval [258]. Near [966]. nearby [800]. neke [49]. Netherlands [713]. netlib [496]. Network [80, 45, 47, 135, 141, 175, 231, 424, 688, 690, 687, 689, 759, 906]. networks [166, 851]. Neuer [772]. Neuere [156]. neural [851]. Newton [200, 216, 225, 262, 511, 790, 19, 27, 79, 85, 244, 253, 332, 527, 550, 682, 683, 792, 793, 794, 791, 803, 868]. Newtonmethode [262]. NJ [55]. no [38]. non [140]. nonbasic [134]. nonconvex [324, 450, 644, 645, 939, 940]. nondegenerate [294]. nondifferentiable [264, 266, 263]. Nonlinear [105, 196, 480, 77, 78, 111, 113, 177, 194, 207, 231, 254, 314, 316, 315, 332, 358, 424, 436, 456, 466, 468, 614, 613, 638, 648, 663, 662, 668, 669, 676, 677, 743, 780, 914]. nonlinearized [177]. nonlinearly [913]. norm [734]. note [54, 254, 564, 836]. Notes [118]. notion [765, 766]. Novel [170]. NP [410, 408]. null [746, 816]. Numerical [20, 67, 204, 257, 480, 829, 903, 51, 130, 384, 387, 457, 560, 654, 725, 768, 796, 830, 912]. numericke [49]. numerique [654, 829]. Numerische [322, 903, 725]. nutshell [781]. NY [144].

OB1 [497]. objective [1, 177, 278, 850, 927, 926, 925, 928]. objectives [112]. observers [767]. obtains [463]. October [144, 170]. oeuvre [787]. Office [258]. one [418, 421, 474, 543, 579, 785]. only [221]. ONR [258]. operational [76]. Operations [81, 347, 291, 549, 673, 869, 696]. Optimal [171, 832, 134, 239, 278, 430, 485, 528, 531, 554, 635, 798, 801, 956, 965]. optimality [134]. Optimierung [156, 261, 262, 423]. Optimierungsaufgaben [226]. Optimierungsmethoden [81]. Optimierungsproblemen [56]. Optimierungsverfahren [693]. optimisation [897]. Optimization [81, 105, 117, 144, 170, 197, 257, 398, 480, 481, 556, 616, 622, 655, 693, 904, 1, 76, 85, 130, 226, 248, 264, 265, 266, 263, 283, 286, 301, 325, 332, 412, 563, 570, 571, 573, 602, 607, 648, 663, 673, 732, 731, 764, 796, 852, 850, 913, 917, 915, 923, 954, 968]. Optimizing [174, 111, 207]. order [99, 112, 172, 452, 455, 547, 539, 558]. ordering [567, 719]. original [260]. Originalproblem [260]. orthant [874]. other [840]. Overseas [759, 422]. overview [298, 741].

Papers [144]. Parallel [352, 353, 650, 720, 18, 19, 133, 175, 271, 501, 607, 642, 663, 675, 872, 897]. parallelization [627]. parallels [617]. parameter [459]. parameterization [1]. parameters [76, 429]. parametric [11, 216, 223, 224, 225, 790]. part [987, 988, 77, 78, 211, 212, 302, 303, 410, 466, 570, 571, 598, 599, 837, 838]. partial [42, 104]. particular [227]. partitioning [686, 749, 748]. passes [774]. Patente [310]. Patents [310, 146, 844]. Path [305, 293, 316, 450, 551, 681, 4, 34, 224, 296, 302, 303, 341, 342, 340, 337, 384, 432, 453, 478, 558, 583, 582, 585, 598, 599, 627, 770, 771, 824, 823, 854, 914, 920, 936, 942, 970, 975, 982, 986, 985]. path-following [936]. paths [314, 715]. Pathways [528]. PC [60]. penalty [195, 361, 364, 368, 370, 712, 706]. perfect [21, 568]. Performance [201, 202, 5, 40, 58, 125, 192, 471, 539, 702, 719, 787, 828, 881]. performances [828]. period [360]. personal [503]. Perspectives [374, 217]. Perspektivy [374]. perturbations [453]. Perturbed [195].

Phase [64, 487, 29, 28, 203, 205, 219, 252, 825, 814, 810, 219]. Pictures [236]. piecewise [143]. pivoting [698]. place [783]. plan [154]. planes [269, 749]. planning [141, 166, 422, 759]. Playing [820]. Point [142, 241, 529, 713, 1, 13, 15, 10, 7, 6, 19, 30, 37, 60, 65, 64, 67, 75, 84, 87, 88, 92, 101, 103, 113, 112, 133, 134, 136, 137, 135, 140, 143, 154, 174, 173, 175, 177, 180, 181, 186, 193, 201, 202, 215, 220, 214, 211, 212, 239, 243, 251, 265, 271, 274, 282, 273, 275, 285, 284, 286, 300, 313, 312, 311, 319, 318, 315, 317, 320, 324, 323, 326, 332, 333, 334, 366, 380, 386, 387, 397, 400, 418, 421, 412, 420, 410, 419, 408, 411, 416, 431, 434, 445, 438, 439, 443, 437, 456, 461, 464, 468, 470, 473, 479, 499, 497, 491, 490, 493, 495, 487, 485, 484, 483, 488, 494, 492, 498, 500, 501, 505]. point [510, 506, 507, 511, 504, 509, 515, 520, 532, 542, 550, 543, 534, 538, 548, 559, 560, 565, 562, 581, 586, 588, 596, 595, 602, 607, 608, 617, 627, 632, 630, 637, 640, 644, 647, 649, 648, 663, 668, 673, 669, 676, 675, 684, 686, 691, 689, 716, 717, 720, 722, 721, 724, 730, 731, 733, 736, 735, 749, 741, 740, 739, 742, 747, 748, 753, 754, 756, 776, 775, 794, 797, 798, 799, 801, 803, 813, 817, 815, 833, 837, 838, 845, 852, 851, 863, 886, 882, 889, 890, 900, 901, 917, 919, 920, 921, 924, 954, 945, 957, 958, 968, 960, 980, 981, 979, 977, 978, 993]. points [314, 820, 935]. polyhedra [69, 209, 210]. polyhedral [212]. polyhedron [603]. polyhedrons [763]. Polynomial [70, 82, 83, 221, 299, 343, 581, 625, 626, 645, 706, 808, 893, 56, 72, 63, 106, 164, 165, 191, 223, 244, 262, 313, 327, 339, 386, 403, 404, 405, 414, 442, 489, 515, 589, 577, 597, 601, 603, 624, 634, 682, 709, 711, 700, 703, 704, 715, 802, 853, 956, 965, 967, 971, 926, 957, 979, 978, 986]. Polynomial-Time [893]. polynomialen [56, 262]. polynomiality [364, 367, 475]. polynomially [922]. polytope [208, 427, 850, 867, 868]. polytopes [1, 66, 947, 941]. polyvector [366]. ponasanje [50]. positive [140, 444]. Potential [947, 29, 39, 42, 159, 221, 213, 228, 227, 274, 296, 295, 297, 316, 325, 336, 340, 356, 357, 362, 389, 394, 439, 446, 474, 479, 512, 580, 594, 646, 652, 684, 707, 698, 702, 818, 814, 868, 937, 942, 938, 946, 944, 935, 955, 948, 966, 961, 959, 303]. pour [472]. Power [413, 601, 772]. Practical [197, 234, 589, 482, 535, 774]. Practice [242, 713, 664]. preconditioned [536, 735, 128]. Preconditioners [248, 538]. prediction [545]. predictor [112, 164, 490, 541, 803, 824]. Preliminary [659]. preprocessing [724]. preserving [475]. price [131, 962]. prices [267]. Pricing [620]. Primal [249, 513, 514, 2, 84, 113, 136, 137, 138, 139, 185, 221, 247, 282, 273, 275, 308, 297, 294, 307, 355, 356, 357, 373, 371, 382, 434, 443, 499, 491, 493, 487, 484, 494, 492, 500, 510, 512, 520, 519, 527, 531, 542, 544, 541, 588, 596, 598, 599, 597, 601, 750, 738, 760, 788, 802, 822, 859, 875, 919, 923, 966, 980, 978, 500]. primer [653]. principales [426]. principle [428, 621, 662, 723]. Principles [227]. Prior [87, 134]. Probabilistic [811, 960]. problem [57, 223, 214, 247, 259, 260, 397, 419, 439, 432, 530, 583, 582, 577, 608, 627, 628, 636, 638, 707, 699, 701, 700, 703, 702, 716, 752, 766, 774, 790, 818, 889, 890, 894, 938, 957]. probleme [654]. problems [13, 12, 14, 46, 47, 52, 56, 62, 64, 75, 85, 99, 101, 129, 135, 143, 160, 163, 164, 172, 178, 191, 201, 202, 216, 225, 226, 231, 237, 300, 291, 313, 314, 316, 315, 323, 325, 326, 336, 335, 386, 391, 420, 410, 408, 424, 431, 433, 440, 438, 441, 435, 436, 444, 446, 442, 437, 448, 450, 459, 486, 515, 563, 574, 566, 567, 572, 570, 571, 568, 573, 578, 581, 589, 600, 595, 632, 631, 630, 644, 647, 651, 649, 645, 646, 652, 648, 654, 676, 688, 687, 704, 749, 751, 754, 762, 773, 840, 846, 847, 854, 858, 860, 856, 859, 861, 862, 872, 906, 967, 968, 961, 973, 981, 977, 975, 983, 985, 988, 993]. procedure [672, 724, 774]. procedures [64, 423, 610, 787, 787]. Proceedings [170, 467, 713]. process [161]. processed [632]. processing [199]. production [360, 509, 747]. program [215, 213, 218, 393, 485, 880]. programmation [654, 787]. Programmazione [148]. Programme [500, 693]. programmes [472, 828]. Programming [55, 80, 128, 125, 148, 196, 241, 398, 402, 467, 529, 523, 713, 728, 9, 13, 4, 12, 14, 11, 15, 10, 7, 8, 20, 22, 29, 41, 43, 36, 39, 28, 34, 23, 38, 42, 31, 37, 24, 26, 33, 27, 53, 51, 48, 56, 57, 62, 73, 79, 77, 78, 82, 83, 84, 86, 90, 88, 92, 93, 95, 97, 102, 100, 99, 101, 103, 108, 115, 119, 118, 121, 127, 131, 136, 140, 145, 152, 153, 154, 155, 156, 160, 165, 167, 168, 169, 171, 172, 173, 175, 176, 177, 180, 184, 185, 186, 189, 188, 187, 190, 193, 194, 198, 199, 201, 202, 205, 204, 206, 217, 223, 220, 219, 221, 211, 222, 228, 230, 227]. programming [229, 235, 236, 237, 239, 240, 243, 244, 246, 252, 254, 255, 256, 253, 249, 250, 261, 262, 270, 274, 282, 273, 275, 276, 285, 284, 288, 296, 295, 308, 287, 297, 304, 305, 306, 293, 300, 299, 292, 291, 302, 303, 298, 309, 313, 312, 311, 317, 320, 324, 323, 333, 341, 343, 334, 336, 335, 340, 338, 339, 344, 348, 350, 354, 355, 356, 358, 361, 363, 364, 365, 368, 369, 370, 367, 372, 374, 382, 379, 380, 423, 391, 388, 394, 395, 396, 399, 400, 403, 418, 421, 407, 404, 405, 411, 414, 415, 416, 428, 434, 443, 432, 450, 449, 451, 452, 457, 458, 454, 456, 455, 459, 453, 460, 461, 464, 465, 466, 478, 479, 490, 487, 488]. programming [486, 494, 492, 506, 507, 511, 504, 509, 512, 513, 514, 517, 518, 520, 519, 528, 532, 522, 524, 526, 525, 530, 549, 551, 534, 538, 560, 564, 566, 565, 569, 572, 562, 576, 580, 584, 579, 585, 587, 586, 588, 590, 592, 594, 596, 598, 599, 600, 601, 593, 609, 615, 611, 612, 614, 613, 619, 620, 617, 621, 624, 625, 626, 629, 634, 636, 637, 638, 639, 641, 642, 643, 650, 646, 653, 654, 656, 665, 661, 662, 666, 667, 668, 673, 669, 674, 678, 677, 679, 680, 682, 681, 684, 689, 685, 694, 695, 696, 697, 707, 709, 711, 710, 712, 708, 699, 701, 700, 703, 702, 704, 705, 706, 718, 727, 729, 743, 740, 739]. programming [742, 747, 751, 754, 755, 756, 758, 760, 762, 761, 763, 765, 770, 773, 780, 788, 792, 793, 794, 791, 795, 798, 800, 802, 801, 819, 813, 817, 821, 822, 825, 827, 814, 808, 826, 823, 811, 807, 806, 815, 812, 810, 816, 833, 835, 836, 837, 838, 840, 841, 844, 843, 845, 853, 858, 864, 860, 857, 856, 859, 861, 862, 863, 865, 867, 869, 872, 873, 888, 892, 895, 898, 899, 900, 901, 902, 905, 906, 907, 908, 909, 910, 911, 919, 920, 922, 929, 937, 942, 943, 950, 951, 956, 927, 965, 971, 972, 945, 946, 939, 926, 949, 958, 962, 940, 955, 925, 930, 966, 963, 969, 960, 928, 964, 933, 932, 959, 980, 979, 976, 977, 975, 978, 983, 986, 985, 987, 988, 990]. programovani [374]. Programs [693, 89, 104, 113, 114, 133, 208, 209, 232, 251, 267, 278, 277, 294, 307, 349, 373, 371, 381, 376, 378, 375, 377, 384, 385, 468, 472, 498, 500, 501, 552, 550, 545, 547, 554, 558, 606, 633, 635, 733, 736, 734, 752, 769, 768, 771, 775, 784, 828, 877, 883, 896, 924, 982, 989]. Progress [30, 529, 526]. proiettivo [148]. Projected [85, 818, 221, 253]. Projection [86, 839, 131, 288, 287, 289, 292, 395, 611, 787, 962, 925, 928]. projections [25, 90, 129, 235, 284, 415, 657, 659, 737, 746, 761, 866, 894, 929]. Projective [148, 214, 210, 211, 212, 893, 16, 36, 28, 24, 26, 33, 46, 54, 57, 77, 108, 114, 167, 168, 169, 176, 203, 205, 204, 208, 209, 246, 253, 264, 269, 266, 270, 268, 282, 273, 281, 391, 388, 393, 473, 512, 564, 569, 593, 638, 639, 694, 697, 708, 755, 779, 819, 825, 827, 826, 805, 810, 834, 835, 894, 895, 919, 920, 921, 951, 927, 972, 991, 992, 195, 466]. proof [17, 91, 367, 639, 853]. properties [864]. property [859, 941, 976]. proposed [846]. Prototyping [107]. PROTRAN [107]. proximal [501, 733, 736, 735]. Punkte [500, 776]. purification [458]. Purposes [234].

Quadratic [143, 862, 43, 34, 57, 83, 99, 101, 112, 160, 173, 198, 250, 274, 275, 293, 323, 339, 381, 384, 391, 399, 400, 479, 552, 549, 599, 601, 606, 625, 627, 644, 647, 650, 651, 649, 645, 646, 666, 667, 684, 743, 760, 826, 883, 943, 971, 972, 945, 939, 968, 940, 933, 932, 980]. quadratically [276, 344, 552, 553, 922, 963, 978]. quasi [332, 885]. quickly [737].

random [419]. range [595]. rank [543, 579]. rate [461, 483, 600, 968]. rates [636]. rational [765]. Ready [509, 747]. real [462, 469]. Realisation [725]. Recovering [131, 956, 965, 962]. recovery [470, 489, 672]. reduced [87, 250, 746, 973]. Reducing [872, 910]. reduction [42, 213, 274, 296, 295, 297, 303, 316, 325, 341, 336, 342, 340, 356, 357, 394, 439, 446, 479, 512, 580, 594, 646, 652, 707, 818, 814, 942, 938, 946, 944, 955, 948, 966, 959]. reference [227]. refinery [360]. region [57, 289, 301, 943]. regression [555]. regular [69]. regularity [38]. regularization [432]. Related [529, 364, 518, 698]. Relation [148, 432]. relationship [569, 753]. relatives [909]. Relaxed [278, 272, 279]. relazione [148]. remark [158]. remarks [159, 758]. Renegar [340]. Renewing [154]. Report [150, 151, 526, 220]. representation [290]. requires [549, 869]. rescaled [874]. rescaling [54, 662, 755]. Research [81, 258, 347, 467, 721, 796, 696]. reservoir [673]. Resolution [358, 654, 472]. resource [409, 876]. response [111]. restart [618]. restarting [493]. restrained [111]. result [17, 887, 938]. Results [329, 58, 68, 156, 217, 255, 288, 354, 416, 457, 463, 556, 571, 830, 834, 964]. review [804]. revised [842]. reweighted [779]. Riemannian [411]. Right [605]. Robust [417]. role [65]. router [640]. routing [889, 890]. rows [942]. rule [698]. runs [329].

safeguarded [35]. satisfiability [397]. scalar [238]. Scale [144, 75, 92, 178, 200, 265, 285, 284, 416, 504, 606, 609, 649, 666, 667, 668, 669, 897]. scaled [29, 39, 472, 818]. scaling [12, 14, 45, 71, 72, 74, 68, 63, 73, 70, 77, 91, 115, 138, 139, 152, 153, 188, 207, 221, 281, 304, 294, 307, 346, 372, 373, 371, 451, 452, 455, 536, 537, 540, 557, 601, 593, 679, 688, 690, 687, 752, 779, 784, 819, 855, 858, 860, 859, 875, 877, 878, 887, 906, 952, 939, 926, 949, 973]. scheduling [75]. scheme [846, 936, 950]. Scheveningen [713]. schneller [226]. Science [147]. Search [306, 991, 992, 18, 44, 334, 460, 487, 569, 575, 610, 629]. searches [946]. second [452, 455]. Section [154]. self [280, 533]. self-correcting [280, 533]. semi [48, 193, 444, 454, 678]. semiinfinte [768]. separable [13, 113, 600, 760]. sequence [40, 122, 578]. Sequential [196]. Serial [501, 18]. series [329, 413, 601, 760]. Set [748, 420, 419, 486, 528, 553, 749, 764, 845, 898]. sets [870, 871]. shadow [131, 962]. Shifted [256, 222, 255, 565, 562]. Short [270, 17, 340]. shrinking [970]. significance [405]. SIMD [19]. simple [85, 97, 290, 365, 393, 853, 868]. simpler [656]. simplesso [148]. Simplex [132, 148, 45, 92, 94, 106, 179, 182, 199, 245, 251, 328, 406, 470, 496, 527, 575, 605, 617, 656, 658, 698, 778, 779, 788, 836, 840, 930, 789]. Simplified [683, 91, 243]. simplifying [952]. simulation [556]. simultaneous [356, 357]. since [30]. Single [205, 203, 252]. six [692]. Sixty [692]. sliding [925]. Smale’s [432]. smooth [336, 335, 376, 378, 377, 550, 707, 763]. software [146, 476]. Solution [247, 638, 646, 652, 16, 18, 57, 122, 160, 178, 391, 417, 472, 548, 651, 654, 768, 832, 900]. solutions [140, 239, 319, 430, 444, 536, 965]. solvable [967]. solve [47, 360, 385, 420, 448, 775]. solved [595]. solver [880]. Solving [19, 89, 135, 226, 250, 326, 388, 574, 567, 570, 571, 568, 573, 749, 989, 56, 60, 62, 75, 87, 106, 114, 133, 191, 201, 202, 215, 217, 213, 218, 232, 245, 259, 274, 277, 291, 349, 378, 384, 393, 450, 479, 500, 530, 550, 545, 547, 558, 633, 648, 695, 764, 769, 774, 846, 889, 890, 968, 973, 975, 982, 984, 321, 847]. Some [68, 84, 116, 159, 345, 405, 617, 632, 754, 758, 840, 902, 238, 283, 318, 325, 326, 456, 534, 566, 572, 645, 854, 247]. SOR [717]. sources [191]. space [746]. Sparse [235, 61, 238, 635, 884]. sparsity [805]. special [120, 806]. specialization [47]. speed [238, 474]. Speeding [401, 873, 540]. Splitting [285, 884]. spreadsheet [59, 348]. squares [178, 201, 202, 643, 779]. stability [51]. stabilnosti [49]. standard [35, 36, 26, 237, 385, 856, 988]. start [215, 217, 213, 218]. Starting [493, 671]. state [139, 738, 740, 739]. stationary [957]. status [220]. Steepest [95, 121, 429, 723, 818]. step [19, 43, 34, 32, 93, 296, 308, 304, 294, 341, 342, 340, 357, 431, 434, 478, 583, 584, 587, 588, 822, 843]. steps [41, 73, 270, 295, 302, 303, 343, 392, 475, 671, 712]. stirs [757]. Stochastic [188, 89, 90, 88, 498, 607]. Stopping [239, 671]. Strategies [914, 898]. strategy [845]. strengthened [25, 389]. Strict [36]. strictly [223, 224, 319]. Strongly [313, 489]. structure [132, 806]. Structured [559, 133, 730, 896]. structures [8, 120]. Studies [330]. study [6, 92, 181, 204, 471, 863]. subject [109]. subspaces [174]. Successful [591]. successive [668]. suggested [561]. suggestions [109]. suitable [678]. summary [438]. Summer [467]. SUMT [34, 195]. Super [977, 846, 847]. supercomputer [451, 499, 491, 722]. Supercomputers [124, 178, 506]. superfast [636]. superieurement [828]. superlinear [386, 544, 980, 981]. superlinearly [214, 519, 802, 969, 979]. surface [111]. survey [167, 169, 334, 556]. Switching [527]. Symbolic [635]. Symmetric [885, 886]. Symposium [518, 713]. System [55, 321, 16, 76, 141, 201, 202, 212, 245, 417, 837, 838, 128, 127, 125, 126, 351, 352, 353, 503, 881]. systems [106, 111, 248, 250, 536, 765, 795, 886, 884].

T [55, 128, 127, 125, 126, 233, 321, 351, 352, 353, 881, 774]. taille [472]. task [98]. Teaching [849]. Technique [105, 111, 605, 798, 801]. Techniques [196, 8, 67, 269, 619, 658, 717, 719]. technology [141]. telecommunications [422]. term [459]. termination [544]. test [486, 832]. tests [3, 239, 774]. their [391, 417, 539, 776, 909]. theorem [445, 389]. Theoretical [222, 218, 434, 535]. Theory [242, 370, 713, 728, 211, 768, 987, 570, 664]. Third [172]. threat [182]. Time [893, 72, 63, 70, 106, 164, 165, 191, 223, 221, 327, 343, 403, 404, 405, 414, 442, 489, 515, 581, 589, 577, 601, 603, 626, 645, 682, 700, 703, 704, 706, 715, 853, 967, 971, 957, 979]. Tracing [715]. trajectories [12, 14, 77, 102, 441, 453, 594, 593, 908, 909]. trajectory [699, 701, 700, 703, 702, 704, 912]. transform [78, 466]. transformation [46, 320, 791, 975]. transformations [208, 214, 210, 211, 212, 951]. transmission [141]. transportation [191, 581, 716, 973]. transputers [175]. Treatment [988]. Trends [732, 363]. Trust [289, 57, 301, 943]. Two [566]. type [106, 423, 413, 600, 623, 694, 717, 751, 868, 924].

U.S. [154]. unbounded [816]. Unconstrained [196, 187]. underlying [411]. unified [246, 438, 437, 512, 742, 895]. uniform [440]. unifying [132]. unimodular [312]. University [144]. unknown [278]. unrestricted [566, 572]. unsolvable [321]. unstandard [984]. updates [203, 297, 543]. updating [42, 104, 564, 579, 719]. upper [359, 828]. USA [55, 144, 467]. use [343, 509, 537, 706, 747, 897]. Using [75, 88, 103, 186, 267, 425, 59, 84, 107, 120, 133, 139, 326, 329, 415, 476, 485, 536, 548, 574, 564, 565, 567, 570, 571, 568, 573, 578, 633, 646, 652, 658, 668, 735, 821, 873, 898, 900]. utilisant [831]. utilization [831].

v [374]. Validation [829, 833]. value [278, 555, 774]. Varetz [170]. variable [155, 865]. variables [104, 134, 181, 243, 430, 505, 546, 566, 572, 821, 816, 828, 877, 956, 988]. variant [35, 89, 120, 155, 237, 261, 277, 340, 337, 372, 373, 371, 394, 572, 592, 611, 671, 679, 708, 714, 746, 815, 865, 874]. Variante [261]. Varianten [262]. variantes [426]. Variants [262, 477, 535, 5, 6, 278, 413, 566, 569, 633, 805, 816, 918]. variation [62, 385, 521]. variational [327, 854]. variations [195]. varieties [1]. Vector [199, 451, 176, 352, 366]. Verallgemeinerung [261]. Verfahren [156, 500, 776, 423]. Verfahrens [56, 260, 261, 331, 502]. version [72, 61, 63, 272, 279, 280, 362, 533, 786]. versus [205]. vertex [548, 800]. vertices [183]. Very [92]. via [16, 89, 188, 595, 714, 748]. Vial [261]. view [11, 677, 742]. Vol [81]. Vol. [257]. volume [910]. vs. [496].

w [214, 210]. warm [215, 217, 213, 218]. way [448]. weighted [163, 212, 268, 339, 709]. well [868]. where [191]. which [698, 869]. wings [240]. within [470, 575, 745]. Wolfe [815]. workbench [721]. Workshop [144, 692]. worse [145]. worst [32]. WRIP [721].

X [138]. XMP [745].

Yamashita [857]. Yoshise [42]. Yoshise’s [818].

zero [181, 418, 421]. zur [56, 261, 262, 500].

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[916] M. H. Wright. Linear algebra issues in interior methods. Talk held at the Fourth SIAM Conference on Applied Linear Algebra in Minneapolis, MN, USA, AT & T Bell Laboratories, Murray Hill, NJ 07974, USA, September 1991.

[917] S. J. Wright. An interior point algorithm for linearly constrained optimization. Technical Report MCS–P162–0790, Mathematical and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA, July 1990.

[918] D. Xiao. Fractional linear functions and the analysis of variants of Karmarkar’s method. Talk held at the ORSA/TIMS Joint National Meeting in New York, NY, USA, Dept. of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA, October 1989.

[919] D. Xiao and D. Goldfarb. A primal projective interior point method for linear programming. Talk held at the CORS/ORSA/TIMS Joint National Meeting in Vancouver, British Columbia, Canada, Dept. of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA, May 1989.

[920] D. Xiao and D. Goldfarb. A path–following projective interior point method for linear programming. Talk held at the ORSA/TIMS Joint National Meeting in Philadelphia, PA, USA, Dept. of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA, October 1990.

[921] D. Xiao and D. Goldfarb. On the complexity of a class of projective interior point methods. Talk held at the ORSA/TIMS Joint National Meeting in Anaheim, CA, USA, Dept. of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA, November 1991.

[922] H. Yamashita. A polynomially and quadratically convergent method for linear programming. Working Paper, Mathematical Systems Institute, Inc., Tokyo, Japan, 1986.

[923] H. Yamashita. A class of primal dual method for constrained optimization. Working Paper, Mathematical Systems Institute, Inc., Tokyo, Japan, 1991.

[924] E. K. Yang and W. S. Hwang. An interior point method for dynamic Leontief type linear programs. Talk held at the ORSA/TIMS Joint National Meeting in Las Vegas, NV, USA, Institute of Applied Mathematics, Tsing Hua University, Hsinchu, 30043, People Republic of China, May 1990.

[925] Y. Ye. Barrier projection and sliding current objective method for linear programming. Talk held at the 12th Mathematical Programming Symposium, Boston, MA, USA, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, 1985.

[926] Y. Ye. Cutting–objective and scaling methods—a polynomial algorithm for linear programming. Working Paper, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, 1985. Submitted to Mathematical Programming.

[927] Y. Ye. Cutting current–objective method in projective algorithm for linear programming. Working Paper, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, February 1985.

[928] Y. Ye. K–projection and cutting–objective method for linear programming. Talk held at the 12th Mathematical Programming Symposium, Boston, MA, USA, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, 1985.

[929] Y. Ye. A large group of projections for linear programming. Working Paper, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, 1985.

[930] Y. Ye. A ‘build-down’ simplex — Karmarkar method for linear programming. Technical Report, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, 1987.

[931] Y. Ye. Dual approach of Karmarkar’s algorithm and the ellipsoid method. Working Paper, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, 1987.

[932] Y. Ye. Further development on the interior algorithm for convex quadratic programming. Working Paper, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, 1987.

[933] Y. Ye. Interior algorithms for linear, quadratic, and linearly constrained convex programming. PhD thesis, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, 1987.

[934] Y. Ye. Karmarkar’s algorithm and the ellipsoid method. Operations Research Letters, 6:177–182, 1987.

[935] Y. Ye. Bimatrix equilibrium points and potential functions. Technical Report, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1988.

[936] Y. Ye. The ‘build-down’ scheme for path-following algorithms. Technical Report, Integrated Systems Inc., Santa Clara, CA, USA, 1988.

[937] Y. Ye. A class of potential functions for linear programming. Management Science Working Paper 88–13, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1988.

[938] Y. Ye. A further result on the potential reduction algorithm for the P –matrix linear complementarity problem. Working Paper, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1988.

[939] Y. Ye. On an affine scaling algorithm for nonconvex quadratic programming. Technical Report, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, April 1988. Revised July 1989.

[940] Y. Ye. On the interior algorithms for nonconvex quadratic programming. Manuscript, Integrated Systems, Inc., Santa Clara, CA, USA, 1988.

[941] Y. Ye. A combinatorial property of analytic centers of polytopes. Technical Report, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1989.

[942] Y. Ye. Eliminating columns and rows in potential reduction and path– following algorithms for linear programming. Working Paper Series 89–7, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1989.

[943] Y. Ye. An extension of Karmarkar’s algorithm and the trust region method for quadratic programming. In N. Megiddo, editor, Progress in Mathematical Programming : Interior Point and Related Methods, pages 49–64. Springer-Verlag, New York, NY, USA, 1989.

[944] Y. Ye. Further developments in potential reduction algorithm. Technical Report, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1989.

[945] Y. Ye. Interior point algorithms for quadratic programming. Working Paper Series 89–29, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1989. To appear in S. Kumar, editor, Recent Developments in Mathematical Programming, Gordon & Beach Scientific Publishers, 1991.

[946] Y. Ye. Line searches in potential reduction algorithm for linear programming. Manuscript, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1989.

[947] Y. Ye. Potential functions and polytopes. Talk held at the ORSA/TIMS Joint National Meeting in New York, NY, USA, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, October 1989.

[948] Y. Ye. A potential reduction algorithm allowing column generation. Working Paper, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1989.

[949] Y. Ye. Anticipated behavior of affine scaling algorithms for linear programming. Technical Report, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1990.

[950] Y. Ye. A ‘build-down’ scheme for linear programming. Mathematical Programming, 46:61–72, 1990.

[951] Y. Ye. A class of projective transformations for linear programming. SIAM Journal on Computing, 19:457–466, 1990.

[952] Y. Ye. Comparative analysis of affine scaling algorithms based on simplifying assumptions. Technical Report, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, January 1990. Revised November 1990. To appear in Mathematical Programming, Series B, 1991.

[953] Y. Ye. Complexity analysis on Karmarkar’s algorithm. Manuscript, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1990.

[954] Y. Ye. Interior point algorithms for global optimization. Annals of Operations Research, 25:59–74, 1990.

[955] Y. Ye. An O(n^{3}L) potential reduction algorithm for linear programming. In J. C. Lagarias and M. J. Todd, editors, Mathematical Developments Arising from Linear Programming : Proceedings of a Joint Summer Research Conference held at Bowdoin College, Brunswick, Maine, USA, June/July 1988, volume 114 of Contemporary Mathematics, pages 77– 88. American Mathematical Society, Providence, Rhode Island, USA, 1990.

[956] Y. Ye. Recovering optimal basic variables in Karmarkar’s polynomial algorithm for linear programming. Mathematics of Operations Research, 15:564–572, 1990.

[957] Y. Ye. A fully polynomial–time approximation algorithm for computing a stationary point of the generalized linear complementarity problem. Talk held at the ORSA/TIMS Joint National Meeting in Nashville, Tennessee, USA, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, May 1991.

[958] Y. Ye. On the finite convergence of interior–point algorithms for linear programming. Technical Report 91–05, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1991.

[959] Y. Ye. An O(n^{3}L) potential reduction algorithm for linear programming. Mathematical Programming, 50:239–258, 1991.

[960] Y. Ye. Toward probabilistic analysis of interior–point algorithms for linear programming. Working Paper Series 91–13, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1991.

[961] Y. Ye. The potential algorithm for linear complementarity problems. Technical Report, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1998.

[962] Y. Ye and S. S. Chiu. Recovering the shadow price in projection methods for linear programming. Technical Report, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1985.

[963] Y. Ye, O. Güler, R. A. Tapia, and Y. Zhang. A quadratically convergent O(\sqrt{n}L)–iteration algorithm for linear programming. Technical Report TR–91–26, Dept. of Mathematical Sciences, Rice University, Houston, TX 77251, USA, August 1991.

[964] Y. Ye and J. A. Kaliski. Further results on build–up approaches for linear programming. Talk held at the ORSA/TIMS Joint National Meeting in, Anaheim, CA, USA, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, November 1991.

[965] Y. Ye and M. Kojima. Recovering optimal dual solutions in Karmarkar’s polynomial algorithm for linear programming. Mathematical Programming, 39:305–317, 1987.

[966] Y. Ye, K. O. Kortanek, J. A. Kaliski, and S. Huang. Near–boundary behavior of primal–dual potential reduction algorithms for linear programming. Working Paper, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1990.

[967] Y. Ye and P. M. Pardalos. A class of linear complementarity problems solvable in polynomial time. Linear Algebra and Its Applications, 152:3– 17, 1991.

[968] Y. Ye and F. A. Potra. An interior–point algorithm for solving entropy optimization problems with globally linear and locally quadratic convergence rate. Working Paper Series 90–22, Dept. of Management Science, University of Iowa, Iowa City, IA 52242, USA, 1990.

[969] Y. Ye, R. A. Tapia, and Y. Zhang. A superlinearly convergent O(\sqrt{n}L)– iteration algorithm for linear programming. Technical Report TR–91–22, Dept. of Mathematical Sciences, Rice University, Houston, TX 77251, USA, 1991.

[970] Y. Ye and M. J. Todd. Containing and shrinking ellipsoids in the path– following algorithm. Mathematical Programming, 47:1–10, 1990.

[971] Y. Ye and E. Tse. A polynomial–time algorithm for convex quadratic programming. Working Paper, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, 1986.

[972] Y. Ye and E. Tse. An extension of Karmarkar’s projective algorithm for convex quadratic programming. Mathematical Programming, 44:157– 179, 1989.

[973] Q. J. Yeh. A reduced dual affine scaling algorithm for solving assignment and transportation problems. PhD thesis, Dept. of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA, 1989.

[974] D. Zaijan. A modified Karmarkar’s algorithm. Applied Mathematics—A Journal of Chinese Universities, 3:41–56, 1988.

[975] J. Z. Zhang. The affine transformation method and the path following method for solving linear programming problems. Chinese Journal of Operations Research, 8(2):25–34, 1989. (In Chinese).

[976] S. Zhang. On the convergence property of Iri–Imai’s method for linear programming. Technical Report 8917/A, Economic Institute, Erasmus University, Rotterdam, The Netherlands, 1989.

[977] Y. Zhang. Super–linear convergence of interior point algorithms for a class of mathematical programming problems. Talk held at the ORSA/TIMS Joint National Meeting in Nashville, Tennessee, USA, Dept. of Mathematics & Statistics, University of Maryland Baltimore County, Baltimore, MD 21228, USA, May 1991.

[978] Y. Zhang and R. A. Tapia. A quadratically convergent polynomial primal–dual interior–point algorithm for linear programming. Technical Report TR 90–40, Dept. of Mathematical Sciences, Rice University, Houston, TX 77251, USA, 1990.

[979] Y. Zhang and R. A. Tapia. A polynomial–time and superlinearly convergent interior point algorithm for linear programming. Technical Report, Dept. of Mathematical Sciences, Rice University, Houston, TX 77251, USA, March 1991.

[980] Y. Zhang, R. A. Tapia, and J. E. Dennis. On the superlinear and quadratic convergence of primal–dual interior point linear programming algorithms. Technical Report TR 90–6, Dept. of Mathematical Sciences, Rice University, Houston, TX 77251, USA, January 1990.

[981] Y. Zhang, R. A. Tapia, and F. Potra. On the superlinear convergence of interior point algorithms for a general class of problems. Technical Report TR 90–9, Dept. of Mathematical Sciences, Rice University, Houston, TX 77251, USA, March 1990.

[982] G. Zhao and J. Stoer. Estimating the complexity of path following methods for solving linear programs by curvature integrals. Technical Report 225, Institut für Angewandte Mathematik und Statistik, Universität Würzburg, Am Hubland, D-8700 Würzburg, Germany, 1990.

[983] Q. Zheng. On the complexity of linear programming problems. Chinese Journal of Operations Research, 7(2):1–10, 1988. (In Chinese).

[984] J. Zhou, N. Xu, and W. Chen. Discussion on Karmarkar’s method for solving unstandard model. Journal of Southeastern University (Nanjing, PR China), English Edition, 5(1):38–45, 1989.

[985] J. Zhu. A path following algorithm for a class of convex programming problems. Working Paper Series 90–14, College of Business Administration, University of Iowa, Iowa City, IA 52240, USA, 1990.

[986] J. Zhu and K. O. Kortanek. A polynomial path–following algorithm for linearly constrained convex programming. Talk held at the ORSA/TIMS Joint National Meeting in Philadelphia, PA, USA, College of Business Administration, University of Iowa, Iowa City, IA 52240, USA, October 1990.

[987] K. Zikan and R. W. Cottle. The box method for linear programming, part I : Basic theory. Technical Report, Systems Optimization Laboratory, Dept. of Operations Research, Stanford University, Stanford, CA 94305-4022, USA, 1987.

[988] K. Zikan and R. W. Cottle. The box method for linear programming, part II : Treatment of problems in standard form with explicitly bounded variables. Technical Report, Systems Optimization Laboratory, Dept. of Operations Research, Stanford University, Stanford, CA 94305-4022, USA, 1987.

[989] K. Zikan and R. W. Cottle. Solving linear programs with the box method. Talk held at the ORSA/TIMS Joint National Meeting in Washington, DC, USA, Systems Optimization Laboratory, Dept. of Operations Research, Stanford University, Stanford, CA 94305-4022, USA, April 1988.

[990] U. Zimmermann. On recent developments in linear programming. In K. H. Hoffmann, J. B. Hiriat-Urruty, C. Lemarechal, and J. Zowe, editors, Trends in Mathematical Optimization : Proceedings of the 4th French–German Conference on Optimization in Irsee, Germany, April 1986, volume 84 of International Series of Numerical Mathematics, pages 353–390. Birkhäuser Verlag, Basel, Switzerland, 1988.

[991] U. Zimmermann. Search directions for projective methods. Technical Report, Technische Universität Braunschweig, Institut für Angewandte Mathematik, Abt. für Math. Optimierung, Pockelstr. 14, D-3300 Braunschweig, Germany, February 1989.

[992] U. Zimmermann. Search directions for a class of projective methods. Zeitschrift für Operations Research—Methods and Models of Operations Research, 34:353–379, 1990.

[993] U. Zimmermann and C. Wallacher. An interior point method for flow problems. Talk held at the DGOR–Jahrestagung in Vienna, Austria, Technische Universität Braunschweig, Institut für Angewandte Mathematik, Abt. für Math. Optimierung, Pockelstr. 14, D-3300 Braunschweig, Germany, August 1990.