Convergent Semidefinite Programming Relaxations for Global Bilevel Polynomial Optimization Problems

Jeyakumar, V.; Lasserre, Jean-Bernard; Li, Guoyin; Phạm, Tiến-Sơn

doi:10.1137/15m1017922

Cited by 23 publications

(27 citation statements)

References 37 publications

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“…[19] as well as by Zeng and An [34]. As well as in the following, the optimal value function ϕ(x) is approximated from above.…”

Section: Definition 32 (Mordukhovichmentioning

confidence: 99%

“…Dempe et al [11]), it is closely related to (generalized) semi-infinite programming ((G)SIP) (see Günzel et al. [12], Stein [13], Stein and Still [14], Weber [15]), mathematical programs with equilibrium (complementarity) constraints (Outrata et al [16]), nonsmooth (Dempe et al [3]), set-valued (Dempe and Pilecka [17]) and (mixed-)integer optimization (Audet et al [18]); ideas of semidefinite optimization can be used to solve it (Jeyakumar et al [19]). …”

Section: Introductionmentioning

confidence: 99%

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On the solution of convex bilevel optimization problems

Dempe

Franke

2015

Comput Optim Appl

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An algorithm is presented for solving bilevel optimization problems with fully convex lower level problems. Convergence to a local optimal solution is shown under certain weak assumptions. This algorithm uses the optimal value transformation of the problem. Transformation of the bilevel optimization problem using the FritzJohn necessary optimality conditions applied to the lower level problem is shown to exhibit almost the same difficulties for solving the problem as the use of the KarushKuhn-Tucker conditions.

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“…[19] as well as by Zeng and An [34]. As well as in the following, the optimal value function ϕ(x) is approximated from above.…”

Section: Definition 32 (Mordukhovichmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the solution of convex bilevel optimization problems

Dempe

Franke

2015

Comput Optim Appl

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“…We refer to [18,19] for the recent work in this area. Recently, Jeyakumar, Lasserre, Li and Pham [16] worked on simple bilevel polynomial programs. When the lower level program (1.2) is convex for each fixed x, they transformed (1.1) into a single level polynomial program, by using Fritz John conditions and the multipliers to replace the lower level program, and globally solving it by using Lasserre type relaxations.…”

Section: Introductionmentioning

confidence: 99%

Bilevel Polynomial Programs and Semidefinite Relaxation Methods

Nie¹,

Wang²,

Ye³

2017

SIAM J. Optim.

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Abstract. A bilevel program is an optimization problem whose constraints involve the solution set to another optimization problem parameterized by upper level variables. This paper studies bilevel polynomial programs (BPPs), i.e., all the functions are polynomials. We reformulate BPPs equivalently as semi-infinite polynomial programs (SIPPs), using Fritz John conditions and Jacobian representations. Combining the exchange technique and Lasserre type semidefinite relaxations, we propose a numerical method for solving bilevel polynomial programs. For simple BPPs, we prove the convergence to global optimal solutions. Numerical experiments are presented to show the efficiency of the proposed algorithm.

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“…In the Appendix, we show how a numerically checkable characterization of robust feasible solution can be obtained in the case of ball data uncertainty in the constraints. Related recent work on global bilevel polynomial optimization in the absence of data uncertainty can be found in [20,23].…”

Section: • Characterizations Of Lower-level Robust Solutions and Spectrmentioning

confidence: 99%

“…However, Putinar Positivstellensatz [33] together with Lasserre type semidefinite relaxations allows us to characterize global optimal solutions and find the global optimal value of a non-convex optimization involving polynomials. The reader is referred to [21][22][23][25][26][27] for related recent work on single level convex and non-convex polynomial optimization in the literature.…”

Section: Introductionmentioning

confidence: 99%

Finding Robust Global Optimal Values of Bilevel Polynomial Programs with Uncertain Linear Constraints

Chương

Jeyakumar

2017

J Optim Theory Appl

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This paper studies, for the first time, a bilevel polynomial program whose constraints involve uncertain linear constraints and another uncertain linear optimization problem. In the case of box data uncertainty, we present a sum of squares polynomial characterization of a global solution of its robust counterpart where the constraints are enforced for all realizations of the uncertainties within the prescribed uncertainty sets. By characterizing a solution of the robust counterpart of the lower-level uncertain linear program under spectrahedral uncertainty using a new generalization of Farkas' lemma, we reformulate the robust bilevel program as a single level non-convex polynomial optimization problem. We then characterize a global solution of the single level polynomial program by employing Putinar's Positivstellensatz of algebraic geometry under coercivity of the polynomial objective function. Consequently, we show that the robust global optimal value of the bilevel program is the limit of a sequence of values of Lasserre-type hierarchy of semidefinite linear programming relaxations. Numerical examples are given to show how the robust optimal value of the bilevel program can be calculated by solving semidefinite programming problems using the Matlab toolbox YALMIP.

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Convergent Semidefinite Programming Relaxations for Global Bilevel Polynomial Optimization Problems

Cited by 23 publications

References 37 publications

On the solution of convex bilevel optimization problems

On the solution of convex bilevel optimization problems

Bilevel Polynomial Programs and Semidefinite Relaxation Methods

Finding Robust Global Optimal Values of Bilevel Polynomial Programs with Uncertain Linear Constraints

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