Generalized semi-infinite programming: A tutorial

Vázquez, F. Guerra; Rückmann, Jan-J.; Stein, Oliver; Still, Georg J.

doi:10.1016/j.cam.2007.02.012

Cited by 68 publications

(17 citation statements)

References 73 publications

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“…Here, for the sake of explanation we restrict our considerations to the particular case of SIP (see also the discussion in Section 4). Secondly, in Example 3.3.13 (from [40,111]) Case II turns out to be stable under arbitrary small C 1 -perturbations of defining functions. For sufficiently small ε > 0 we perturb the functions g 0 , g 1 , g 2 as follows: …”

Section: Reduction Under Nsramentioning

confidence: 91%

“…In case of a constant mapping Y (·) = Y , we refer to semi-infinite optimization problems (SIP). We present the well-known applications of GSIP in the area of Chebyshev approximation, design centering and robust optimization from a survey [40]. Further, we give some examples of GSIP which illustrate two main new features of GSIP (in addition to SIP):…”

Section: Applications and Examplesmentioning

confidence: 99%

“…Moreover, the local nonclosedness of the feasible set M is stable (e.g. [40]). Therefore, one considers the topological closure M of M instead.…”

Section: Assumption B the Mappingmentioning

confidence: 99%

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Topological Aspects of Nonsmooth Optimization

Shikhman

2012

Springer Optimization and Its Applications

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The goal of this paper is to compare alternative stationarity notions in structured nonsmooth optimization (SNO). Here, nonsmoothness is caused by complementarity, vanishing, orthogonality type, switching, or disjunctive constraints. On one side, we consider geometrically motivated notions of N -, N -, and N -stationarity in terms of Fréchet, Mordukhovich, and Clarke normal cones to the feasible set, respectively. On the other side, we advocate the notion of topologically relevant T-stationarity, which adequately captures the global structure of SNO. Our main findings say that (a) N -stationary points include all local minimizers; (b) Nstationary points, which are not N -stationary, correspond to the singular saddle points of first order; (c) T-stationary points, which are not N -stationary, correspond to the regular saddle points of first order; (d) N -stationary points, which are not T-stationary, are irrelevant for optimization purposes. Overall, a hierarchy of stationarity notions for SNO is established.

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Section: Reduction Under Nsramentioning

confidence: 91%

Section: Applications and Examplesmentioning

confidence: 99%

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Topological Aspects of Nonsmooth Optimization

Shikhman

2012

Springer Optimization and Its Applications

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“…It is worth mentioning that there exists a wide range of applications for this problem class, like, e.g., Chebyshev and reverse Chebyshev approximation, Design centering, or Robust optimization. For a nice overview we refer to [4]. Our results can directly be applied to GSIP.…”

Section: Application To Gsipmentioning

confidence: 89%

On Topological Properties of Min-Max Functions

2010

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We examine the topological structure of the upper-level set M max given by a min-max function ϕ. It is motivated by recent progress in Generalized Semi-Infinite Programming (GSIP). Generically, M max is proven to be the topological closure of the GSIP feasible set (see Guerra-Vázquez et al. 2009; Günzel et al., Cent Eur J Oper Res 15(3):271-280, 2007). We formulate two assumptions (Compactness Condition CC and Sym-MFCQ) which imply that M max is a Lipschitz manifold (with boundary). The Compactness Condition is shown to be stable under C 0 -perturbations of the defining functions of ϕ. Sym-MFCQ can be seen as a constraint qualification in terms of Clarke's subdifferential of the min-max function ϕ. Moreover, Sym-MFCQ is proven to be generic and stable under C 1 -perturbations of the defining functions which fulfill the Compactness Condition. Finally we apply our results to GSIP and conclude that generically the closure of the GSIP feasible set is a Lipschitz manifold (with boundary).

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“…In particular, for the auxiliary relaxed inner problem we apply classical convexification techniques [36,71] to construct and solve a convex problem. For the auxiliary restricted inner problem, we apply generalized semi-infinite programming techniques [41,69] to reduce it to a finite problem. The resulting problem is nonconvex; hence, we can either solve a convex relaxation or solve it directly to global optimality.…”

Section: Bounding Scheme: Initial Nodementioning

confidence: 99%

Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development

Kleniati

Adjiman

2014

J Glob Optim

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We present a global optimization algorithm, Branch-and-Sandwich, for optimistic bilevel programming problems that satisfy a regularity condition in the inner problem. The functions involved are assumed to be nonconvex and twice continuously differentiable. The proposed approach can be interpreted as the exploration of two solution spaces (corresponding to the inner and the outer problems) using a single branch-and-bound tree. A novel branching scheme is developed such that classical branch-and-bound is applied to both spaces without violating the hierarchy in the decisions and the requirement for (global) optimality in the inner problem. To achieve this, the well-known features of branch-and-bound algorithms are customized appropriately. For instance, two pairs of lower and upper bounds are computed: one for the outer optimal objective value and the other for the inner value function. The proposed bounding problems do not grow in size during the algorithm and are obtained from the corresponding problems at the parent node.Keywords Bilevel programming · Nonconvex inner problem · Branch and bound

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Generalized semi-infinite programming: A tutorial

Cited by 68 publications

References 73 publications

Topological Aspects of Nonsmooth Optimization

Topological Aspects of Nonsmooth Optimization

On Topological Properties of Min-Max Functions

Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development

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