Generalized Semi-infinite Programming Research Articles

Generalized corner optimal solution for LSIP: existence and numerical computation

In this paper, we consider general linear semi-infinite programming (LSIP) problems and study the existence and computation of optimal solutions at special generalized corner points called generalized ladder points (glp). We develop conditions, including an equivalent condition, under which glp optimal solutions exist. These results are fundamentally important to the ladder method for LSIP, which finds an optimal solution at a glp in the feasible region. For problems that do not have glp optimal solutions, we propose the addition of special artificial constraints to the constraint system of the problem to create a glp optimal solution. We present a ladder algorithm based on the maximum violation rule and an artificial ladder technique. Convergence results are provided with the support of some numerical tests.

Sequential algorithms for structural design optimization under tolerance conditions

A deterministic optimization usually ignores the effects of uncertainties in design variables or design parameters on the constraints. In practical applications, it is required that the optimum solution can endure some tolerance so that the constraints are still satisfied when the solution undergoes variations within the tolerance range. An optimization problem under tolerance conditions is formulated in this article. It is a kind of robust design and a special case of a generalized semi-infinite programming (GSIP) problem. To overcome the deficiency of directly solving the double loop optimization, two sequential algorithms are then proposed for obtaining the solution, i.e. the double loop optimization is solved by a sequence of cycles. In each cycle a deterministic optimization and a worst case analysis are performed in succession. In sequential algorithm 1 (SA1), a shifting factor is introduced to adjust the feasible region in the next cycle, while in sequential algorithm 2 (SA2), the shifting factor is replaced by a shifting vector. Several examples are presented to demonstrate the efficiency of the proposed methods. An optimal design result based on the presented method can endure certain variation of design variables without violating the constraints. For GSIP, it is shown that SA1 can obtain a solution with equivalent accuracy and efficiency to a local reduction method (LRM). Nevertheless, the LRM is not applicable to the tolerance design problem studied in this article.

ON FIRST-ORDER NECESSARY CONDITIONS FOR OPTIMALITY OF REGULAR GENERALIZED SEMI-INFINITE PROGRAMMING

This paper is concerned with the optimality for generalized semi-infinite programming (GSIP) with nondifferentiable and nonconvex (but being regular in Clarke sense) constraint functions. The objective function is only locally Lipschitz. We consider a lower level constraint qualification which is based on the Clarke subdifferential. This constraint qualification is a natural extension of Mangasarian–Fromovitz one to the differentiable GSIP. The main results are Fritz-John type necessary conditions for optimal solutions.

Semi-infinite programming method for optimal power flow with transient stability and variable clearing time of faults

This paper presents a new nonlinear programming problem arising in the control of power systems, called optimal power flow with transient stability constraint and variable clearing time of faults and abbreviated as OTS-VT. The OTS-VT model is converted into a implicit generalized semi-infinite programming (GSIP) problem. According to the special box structure of the reformulated GSIP, a solution method based on bi-level optimization is proposed. The research in this paper has two contributions. Firstly, it generalizes the OTS study to general optimal power flow with transient stability problems. From the viewpoint of practical applications, the proposed research can improve the decision-making ability in power system operations. Secondly, the reformulation of OTS-VT also provides a new background and a type of GSIP in the research of mathematical problems. Numerical results for two chosen power systems show that the methodology presented in this paper is effective and promising.

General semi-infinite programming: critical point theory

We study General Semi-Infinite Programming (GSIP) from a topological point of view. Under the Symmetric Mangasarian–Fromovitz Constraint Qualification (Sym-MFCQ) two basic theorems from Morse theory (deformation theorem and cell-attachment theorem) are proved. Outside the set of Karush–Kuhn–Tucker (KKT) points, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a KKT level, the topology of the lower level set changes via the attachment of a q-dimensional cell. The dimension q equals the so-called GSIP-index of the (nondegenerate) KKT-point. Here, the Nonsmooth Symmetric Reduction Ansatz (NSRA) allows to perform a local reduction of GSIP to a Disjunctive Optimization Problem. The GSIP-index then coincides with the stationary index from the corresponding Disjunctive Optimization Problem.

Generalized Semi-Infinite Programming: The Nonsmooth Symmetric Reduction Ansatz

The feasible set M in generalized semi-infinite programming (GSIP) need not be closed. Under the so-called symmetric Mangasarian–Fromovitz constraint qualification (Sym-MFCQ), its closure $\overline{M}$ can be described by means of infinitely many inequality constraints of maximum type. In this paper we introduce the nonsmooth symmetric reduction ansatz (NSRA). Under NSRA we prove that the set $\overline{M}$ can locally be described as the feasible set of a so-called disjunctive optimization problem defined by finitely many inequality constraints of maximum type. This also shows the appearance of re-entrant corners in $\overline{M}$. Under Sym-MFCQ all local minimizers of GSIP are KKT points for GSIP. We show that NSRA is generic and stable at all KKT points and that all KKT points are nondegenerate. The concept of (nondegenerate) KKT points as well as a corresponding GSIP-index are introduced in this paper. In particular, a nondegenerate KKT-point is a local minimizer if and only if its GSIP-index vanishes. At local minimizers NSRA coincides with the symmetric reduction ansatz (SRA) as introduced in [H. Günzel, H. Th. Jongen, and O. Stein, Optim. Lett., 2 (2008), pp. 415–424]. In comparison with SRA, the main new issue in NSRA is the following: At KKT points different from local minimizers, the Lagrange polytope at the lower level generically need not be a singleton anymore. In fact, it will be a full-dimensional simplex. This fact is crucial to provide the above-mentioned local reduction to a disjunctive optimization problem. Finally, we establish a local cell-attachment theorem which will be the basis for the development of a global critical point theory for GSIP.

On Topological Properties of Min-Max Functions

We examine the topological structure of the upper-level set Mmax given by a min-max function φ. It is motivated by recent progress in Generalized Semi-Infinite Programming (GSIP). Generically, Mmax is proven to be the topological closure of the GSIP feasible set (see Guerra-Vazquez et al. 2009; Gunzel 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 Mmax is a Lipschitz manifold (with boundary). The Compactness Condition is shown to be stable under C0-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 C1-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).

General Semi-Infinite Programming: Symmetric Mangasarian–Fromovitz Constraint Qualification and the Closure of the Feasible Set

The feasible set $M$ in general semi-infinite programming (GSIP) need not be closed. This fact is well known. We introduce a natural constraint qualification, called symmetric Mangasarian-Fromovitz constraint qualification (Sym-MFCQ). The Sym-MFCQ is a nontrivial extension of the well-known (extended) MFCQ for the special case of semi-infinite programming (SIP) and disjunctive programming. Under the Sym-MFCQ the closure $\overline{M}$ has an easy and also natural description. As a consequence, we get a description of the interior and boundary of $M$. The Sym-MFCQ is shown to be generic and stable under $C^1$-perturbations of the defining functions. For the latter stability the consideration of the closure of $M$ is essential. We introduce an appropriate notion of Karush-Kuhn-Tucker (KKT) points. We show that local minimizers are KKT points under the Sym-MFCQ.

Randomized Strategies for Probabilistic Solutions of Uncertain Feasibility and Optimization Problems

In this paper, we study two general semi-infinite programming problems by means of a randomized strategy based on statistical learning theory. The sample size results obtained with this approach are generally considered to be very conservative by the control community. The first main contribution of this paper is to demonstrate that this is not necessarily the case. Utilizing as a starting point one-sided results from statistical learning theory, we obtain bounds on the number of required samples that are manageable for ldquoreasonablerdquo values of probabilistic confidence and accuracy. In particular, we show that the number of required samples grows with the accuracy parameter epsiv as 1/epsivln 1/epsiv , and this is a significant improvement when compared to the existing bounds which depend on 1/epsiv2ln 1/epsiv2. Secondly, we present new results for optimization and feasibility problems involving Boolean expressions consisting of polynomials. In this case, when the accuracy parameter is sufficiently small, an explicit bound that only depends on the number of decision variables, and on the confidence and accuracy parameters is presented. For convex optimization problems, we also prove that the required sample size is inversely proportional to the accuracy for fixed confidence. Thirdly, we propose a randomized algorithm that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived.

Application of general semi-infinite programming to lapidary cutting problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interior-point method developed by Stein [O. Stein, Bi-level Strategies in Semi-infinite Programming, Kluwer Academic Publishers, Boston, 2003]. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on real-world data are also presented.

Mathematical contributions to dynamics and optimization of gene-environment networks†

This article contributes to a further introduction of continuous optimization in the field of computational biology which is one of the most challenging and emerging areas of science, in addition to foundations presented and the state-of-the-art displayed in [C.A. Floudas and P.M. Pardalos, eds., Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches, Kluwer Academic Publishers, Boston, 2000]. Based on a summary of earlier works by the coauthors and their colleagues, it refines the model on gene-environment patterns by a problem from generalized semi-infinite programming (GSIP), and characterizes the condition of its structural stability. Furthermore, our paper tries to detect and understand structural frontiers of our methods applied to the recently introduced gene-environment networks and tries to overcome them. Computational biology is interdisciplinary, but it also looks for its mathematical foundations. From data got by DNA microarray experiments, non-linear ordinary differential equations are extracted by the optimization of least-squares errors; then we derive corresponding time-discretized dynamical systems. Using a combinatorial algorithm with polyhedra sequences we can detect the regions of parametric stability, contributing to a testing the goodness of data fitting of the model. To represent and interpret the dynamics, certain matrices, genetic networks and, more generally, gene-environment networks serve. Here, we consider n genes in possible dependence with m special environmental factors and a cumulative one. These networks are subject of discrete mathematical questions, but very large structures, such that we need to simplify them. This is undertaken in a careful optimization with constraints, aiming at a balanced connectedness, incorporates any type of a priori knowledge or request and should be done carefully enough to be robust against disturbation by the environment. In this way, we take into account attacks on the network, knockout phenomena and catastrophies, but also changes in lifestyle and effects of education as far as they can approximately be quantified. We characterize the structural stability of the GSIP problem against perturbations like changes between data series or due to outliers. We give explanations on the numerics and the use of splines. This study is an attempt to demonstrate some beauty and applicabilty of continuous optimization which might together one day give a support in health care, food engineering, biomedicine and -technology, including elements of bioenergy and biomaterials. †Dedicated to Prof. Dr H. Th. Jongen on the occasion of his 60th birthday in admiration of his scientific work.

Generalized semi-infinite programming: A tutorial

This tutorial presents an introduction to generalized semi-infinite programming (GSIP) which in recent years became a vivid field of active research in mathematical programming. A GSIP problem is characterized by an infinite number of inequality constraints, and the corresponding index set depends additionally on the decision variables. There exist a wide range of applications which give rise to GSIP models; some of them are discussed in the present paper. Furthermore, geometric and topological properties of the feasible set and, in particular, the difference to the standard semi-infinite case are analyzed. By using first-order approximations of the feasible set corresponding constraint qualifications are developed. Then, necessary and sufficient first- and second-order optimality conditions are presented where directional differentiability properties of the optimal value function of the so-called lower level problem are used. Finally, an overview of numerical methods is given.

Extensions of the Kuhn--Tucker Constraint Qualification to Generalized Semi-infinite Programming

This paper deals with the class of generalized semi-infinite programming problems (GSIPs) in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be continuously differentiable. We introduce two extensions of the Kuhn--Tucker constraint qualification (which is based on the existence of a tangential continuously differentiable arc) to the class of GSIPs, prove a corresponding Karush--Kuhn--Tucker theorem, and discuss its assumptions. Finally, we present several examples which illustrate for the class of GSIPs some interrelations between the considered extensions of the Mangasarian--Fromovitz constraint qualification, the Abadie constraint qualification, and the Kuhn--Tucker constraint qualification.

On Constraint Qualifications in Nonsmooth Optimization

We introduce extensions of the Mangasarian-Fromovitz and Abadie constraint qualifications to nonsmooth optimization problems with feasibility given by means of lower-level sets. We do not assume directional differentiability, but only upper semicontinuity of the defining functions. By deriving and reviewing primal first-order optimality conditions for nonsmooth problems, we motivate the formulations of the constraint qualifications. Then, we study their interrelation, and we show how they are related to the Slater condition for nonsmooth convex problems, to nonsmooth reverse-convex problems, to the stability of parametric feasible set mappings, and to alternative theorems for the derivation of dual first-order optimality conditions. In the literature on general semi-infinite programming problems, a number of formally different extensions of the Mangasarian-Fromovitz constraint qualification have been introduced recently under different structural assumptions. We show that all these extensions are unified by the constraint qualification presented here.