Bi-level Strategies in Semi-infinite ProgrammingSpringer Science & Business Media, 2003. gada 31. aug. - 202 lappuses Semi-infinite optimization is a vivid field of active research. Recently semi infinite optimization in a general form has attracted a lot of attention, not only because of its surprising structural aspects, but also due to the large number of applications which can be formulated as general semi-infinite programs. The aim of this book is to highlight structural aspects of general semi-infinite programming, to formulate optimality conditions which take this structure into account, and to give a conceptually new solution method. In fact, under certain assumptions general semi-infinite programs can be solved efficiently when their bi-Ievel structure is exploited appropriately. After a brief introduction with some historical background in Chapter 1 we be gin our presentation by a motivation for the appearance of standard and general semi-infinite optimization problems in applications. Chapter 2 lists a number of problems from engineering and economics which give rise to semi-infinite models, including (reverse) Chebyshev approximation, minimax problems, ro bust optimization, design centering, defect minimization problems for operator equations, and disjunctive programming. |
Saturs
INTRODUCTION | 5 |
11 Standard semiinfinite programming | 6 |
12 General semiinfinite programming | 8 |
13 The misconception about the generality of GSIP | 10 |
14 Development to a field of active research | 12 |
EXAMPLES AND APPLICATIONS | 15 |
21 Chebyshev and reverse Chebyshev approximation | 16 |
22 Minimax problems | 18 |
43 Dual first order optimality conditions | 120 |
431 The standard semiinfinite case | 122 |
432 The completely convex case | 124 |
433 The convex case | 127 |
434 The C² case with Reduction Ansatz | 130 |
435 The C¹ case | 132 |
44 Second order optimality conditions | 146 |
BILEVEL METHODS FOR GSIP | 149 |
23 Robust optimization | 19 |
24 Design centering | 22 |
26 Disjunctive programming | 26 |
TOPOLOGICAL STRUCTURE OF THE FEASIBLE SET | 29 |
311 A projection formula | 31 |
312 A bilevel formula and semicontinuity properties | 35 |
313 A setvalued mapping formula | 45 |
314 The local structure of M | 46 |
315 The completely convex case | 48 |
32 Index set mappings with functional constraints | 50 |
322 The linear case | 51 |
323 The C¹ case | 64 |
324 The C² case | 66 |
325 Genericity results | 70 |
OPTIMALITY CONDITIONS | 89 |
42 First order approximations of the feasible set | 94 |
421 General constraint qualifications | 95 |
422 Descriptions of the linearization cones | 100 |
423 Degenerate index sets | 112 |
51 Reformulations of GSIP | 150 |
512 The MPEC reformulation of GSIP | 152 |
513 A regularization of MPEC by NCP functions | 153 |
514 The regularized Stackelberg game | 156 |
52 Convergence results for a bilevel method | 158 |
521 A parametric reduction lemma | 159 |
522 Convergence of global solutions | 161 |
523 Convergence of Fritz John points | 162 |
524 Quadratic convergence of the optimal values | 166 |
525 An outer approximation property | 167 |
53 Other bilevel approaches and generalizations | 171 |
COMPUTATIONAL RESULTS | 175 |
61 Design centering in two dimensions | 176 |
62 Design centering in higher dimensions | 181 |
63 Robust optimization | 182 |
64 Optimal error bounds for an elliptic operator equation | 185 |
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Bieži izmantoti vārdi un frāzes
active indices assumptions Chebyshev approximation choice y¹ closed set constraint qualification convergence Corollary defining functions denote directional derivatives DxLi(ã EMFCQ holds Example exist yi,k feasible boundary point feasible set following assertions hold formulation Fritz John gi(x implies index set index set mapping inner semi-continuous Io(x Lemma Li(x LICQ linearization cone linearization cone satisfies linearly independent locally bounded lower level problems lower semi-continuous matrix method MFCQ MGSIP minimizer of GSIP minimizer of order NCP function neighborhood non-empty and compact non-singular numerical optimal value functions order optimality conditions outer semi-continuous parametric polytope Proof Proposition result robust optimization Rückmann Section semi-infinite optimization problem sequence set Y(x set-valued mapping Slater point Slater's condition Stackelberg game standard semi-infinite optimization standard semi-infinite programming sufficient condition Theorem topological topological interior upper semi-continuous ve(x vector y¹‚k Yi(x Σ Σ
Atsauces uz šo grāmatu
Variational Analysis and Generalized Differentiation I: Basic Theory Boris S. Mordukhovich Priekšskatījums nav pieejams - 2012 |