Smoothing by mollifiers. Part II: nonlinear optimization

This article complements the paper (Jongen, Stein, Smoothing by mollifers part I: semi-infinite optimization J Glob Optim doi:), where we showed that a compact feasible set of a standard semi-infinite optimization problem can be approximated arbitrarily well by a level set of a single smooth function with certain regularity properties. In the special case of nonlinear programming this function is constructed as the mollification of the finite min-function which describes the feasible set. In the present article we treat the correspondences between Karush–Kuhn–Tucker points of the original and the smoothed problem, and between their associated Morse indices.      

How to solve a semi-infinite optimization problem

After an introduction to main ideas of semi-infinite optimization, this article surveys recent developments in theory and numerical methods for standard and generalized semi-infinite optimization problems. Particular attention is paid to connections with mathematical programs with complementarity constraints, lower level Wolfe duality, semi-smooth approaches, as well as branch and bound techniques in adaptive convexification procedures. A section on recent genericity results includes a discussion of the symmetry effect in generalized semi-infinite optimization.     

On duality in multiobjective semi-infinite optimization

Abstract

In this paper, we consider multiobjective semi-infinite optimization problems which are defined in a finite-dimensional space by finitely many objective functions and infinitely many inequality constraints. We present duality results both for the convex and nonconvex case. In particular, we show weak, strong and converse duality with respect to both efficiency and weak efficiency. Moreover, the property of being a locally properly efficient point plays a crucial role in the nonconvex case.     

Unified steerable phase I-phase II method of feasible directions for semi-infinite optimization

In this paper, we complete a cycle in the construction of methods of feasible directions for solving semi-infinite constrained optimization problems. Earlier phase I-phase II methods of feasible directions used one search direction rule in all of ⁿ with two stepsize rules, one for feasible points and one for infeasible points. The algorithm presented in this paper uses both a single search direction rule and a single stepsize rule in all of ⁿ . In addition, the new algorithm incorporates a steering parameter which can be used to control the speed with which feasibility is achieved. The new algorithm is simpler to analyze and performs somewhat better than existing, first order, phase I-phase II methods. The new algorithm is globally convergent, with linear rate.The research reported herein was sponsored in part by the National Science Foundation Grant ECS-8713334, the Air Force Office of Scientific Research Contract AFOSR-86-0116, and the State of California MICRO Program Grant 532410-19900.The authors would like to thank Dr. J. Higgins for providing the C-code of Algorithm 3.1.     

Density of stable convex semi-infinite vector optimization problems

In this paper, we show that every convex semi-infinite vector optimization (CSVO for brevity) problem can be arbitrarily approximated by stable CSVO problems, i.e., the set of all stable CSVO problems (the weak solution map is continuous or the solution map is upper semicontinuous) is dense in the set of all CSVO problems with the given topology.     

An approximation of feasible sets in semi-infinite optimization

The paper deals with the feasible setM of a semi-infinite optimization problem, i.e.M is a subset of the finite-dimensional Euclidean space and is basically defined by infinitely many inequality constraints. Assuming that the extended Mangasarian-Fromovitz constraint qualification holds at all points fromM, it is shown that the quadratic distance function with respect toM is continuously differentiable on an open neighborhood ofM. If, in addition,M is compact, then the set , which is described by this quadratic distance function, is shown to be an appropriate approximation ofM and the setsM and can be topologically identified via a homeomorphism.     

Metric regularity of the feasible set mapping in semi-infinite optimization

For parametric systems of (finitely many) equations and (infinitely many) inequalities the well-known concept of metric regularity is shown to be equivalent to the so-called extended Mangasarian-Fromovitz constraint qualification. By this, a corresponding result obtained by Robinson for finite optimization problems my be transferred to semi-infinite optimization. For the proof a local epigraph representation of the constraint set is mainly used.     

Isolated calmness of solution mappings in convex semi-infinite optimization

This paper is concerned with isolated calmness of the solution mapping of a parameterized convex semi-infinite optimization problem subject to canonical perturbations. We provide a sufficient condition for isolated calmness of this mapping. This sufficient condition characterizes the strong uniqueness of minimizers, under the Slater constraint qualification. Moreover, on the assumption that the objective function and the constraints are linear, we show that this condition is also necessary for isolated calmness.     

Radiofrequency ablation planning: An application of semi-infinite modelling techniques

In radiofrequency (RF) ablation a needle-shaped probe is inserted into the patient’s body in order to heat and subsequently destroy the malignant tissue around the needle tip. The determination of the optimal probe position poses an intricate problem, as it requires the modelling of the tumour destruction depending on the attained temperature as well as the incorporation of constraints that prohibit puncturing bones or other risk structures.In this work we present a new optimization procedure that reflects both aspects adequately. We assess tumour destruction by solving the underlying system of partial differential equations using a finite element method. Next we show how the probe’s position and orientation can be optimized by gradient-based methods. Ensuring that risk structures are not harmed by the probe is easily modelled using semi-infinite constraints in the optimization problem.Techniques to reduce the semi-infinite problem to a standard nonlinear constrained optimization problem are introduced and demonstrated as a proof-of-concept on real clinical data. The results indicate the high potential of this combination of PDE-based simulation and numerical optimization for RF ablation planning.     

One-parameter families of optimization problems: Equality constraints

In this paper, we introduce generalized critical points and discuss their relationship with other concepts of critical points [resp., stationary points]. Generalized critical points play an important role in parametric optimization. Under generic regularity conditions, we study the set of generalized critical points, in particular, the change of the Morse index. We focus our attention on problems with equality constraints only and provide an indication of how the present theory can be extended to problems with inequality constraints as well.     

Uniform duality in semi-infinite convex optimization

We study infinite sets of convex functional constraints, with possibly a set constraint, under general background hypotheses which require closed functions and a closed set, but otherwise do not require a Slater point. For example, when the set constraint is not present, only the consistency of the conditions is needed. We provide hypotheses, which are necessary as well as sufficient, for the overall set of constraints to have the property that there is no gap in Lagrangean duality for every convex objective function defined on ℝⁿ. The sums considered for our Lagrangean dual are those involving only finitely many nonzero multipliers. In particular, we recover the usual sufficient condition when only finitely many functional constraints are present. We show that a certain compactness condition in function space plays the role of finiteness, when there are an infinite number of functional constraints. The author's research has been partially supported by Grant ECS8001763 of the National Science Foundation.     

Second-Order Smoothing Objective Penalty Function for Constrained Optimization Problems

In this article, a novel objective penalty function as well as its second-order smoothing is introduced for constrained optimization problems (COP). It is shown that an optimal solution to the second-order smoothing objective penalty optimization problem is an optimal solution to the original optimization problem under some mild conditions. Based on the second-order smoothing objective penalty function, an algorithm that has better convergence is introduced. Numerical examples illustrate that this algorithm is efficient in solving COP.     

On Logarithmic Smoothing of the Maximum Function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation g of f itself, where >0 denotes the approximation parameter. The one-parametric family g converges – relative to a compact subset – uniformly to the function f as tends to zero. Under nondegeneracy assumptions we show that the stationary points of g and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x() of stationary points for g , where each x() converges to the corresponding stationary point of f as tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve x().     

Solving generalized semi-infinite programs by reduction to simpler problems

The article intends to give a unifying treatment of different approaches to solve generalized semi-infinite programs by transformation to simpler problems. In particular dual-, penalty-, discretization-, reduction-, and Karush-Kuhn-Tucker (KKT)-methods are applied to obtain equivalent problems or relaxations of a simpler structure. The relaxations are viewed as a perturbation P _τ of the original problem P, depending on a perturbation parameter τ > 0, and are analyzed by using parametric programming techniques. We give convergence results and results on the rate of convergence for the minimal values and the optimal solutions of P _τ when τ tends toward 0. We review earlier studies and present new ones.     

Constraint qualifications in linear vector semi-infinite optimization

Linear vector semi-infinite optimization deals with the simultaneous minimization of finitely many linear scalar functions subject to infinitely many linear constraints. This paper provides characterizations of the weakly efficient, efficient, properly efficient and strongly efficient points in terms of cones involving the data and Karush–Kuhn–Tucker conditions. The latter characterizations rely on different local and global constraint qualifications. The global constraint qualifications are illustrated on a collection of selected applications.     

Global optimization of truss topology with discrete bar areas—Part I: theory of relaxed problems

This two part paper considers the classical problem of finding a truss design with minimal compliance subject to a given external force and a volume bound. The design variables describe the cross-section areas of the bars. While this problem is well-studied for continuous bar areas, we treat here the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single bar area, i.e., a 0/1-problem. In contrast to heuristic methods considered in other approaches, Part I of the paper together with Part II present an algorithmic framework for the calculation of a global optimizer of the underlying large-scaled mixed integer design problem. This framework is given by a convergent branch-and-bound algorithm which is based on solving a sequence of nonconvex continuous relaxations. The main issue of the paper and of the approach lies in the fact that the relaxed nonlinear optimization problem can be formulated as a quadratic program (QP). Here the paper generalizes and extends the available theory from the literature. Although the Hessian of this QP is indefinite, it is possible to circumvent the non-convexity and to calculate global optimizers. Moreover, the QPs to be treated in the branch-and-bound search tree differ from each other just in the objective function. In Part I we give an introduction to the problem and collect all theory and related proofs for the treatment of the original problem formulation and the continuous relaxed problems. The implementation details and convergence proof of the branch-and-bound methodology and the large-scale numerical examples are presented in Part II.     

Generalized semi-infinite optimization: A first order optimality condition and examples

We consider a generalized semi-infinite optimization problem (GSIP) of the form (GSIP) min{f(x)‖xεM}, where M={x∈ℝⁿ|h_i(x)=0i=l,...m, G(x,y)⩾0, y∈Y(x)} and all appearing functions are continuously differentiable. Furthermore, we assume that the setY(x) is compact for allx under consideration and the set-valued mappingY(.) is upper semi-continuous. The difference with a standard semi-infinite problem lies in thex-dependence of the index setY. We prove a first order necessary optimality condition of Fritz John type without assuming a constraint qualification or any kind of reduction approach. Moreover, we discuss some geometrical properties of the feasible setM. This work was partially supported by the “Deutsche Forschungsgemeinschaft” through the Graduiertenkolleg “Mathematische Optimierung” at the University of Trier.     

Semi-infinite optimization: Structure and stability of the feasible set

The problem of the minimization of a functionf: ⁿ under finitely many equality constraints and perhaps infinitely many inequality constraints gives rise to a structural analysis of the feasible setM[H, G]={xⁿ¦H(x)=0,G(x, y)0,yY} with compactY^r. An extension of the well-known Mangasarian-Fromovitz constraint qualification (EMFCQ) is introduced. The main result for compactM[H, G] is the equivalence of the topological stability of the feasible setM[H, G] and the validity of EMFCQ. As a byproduct, we obtain under EMFCQ that the feasible set admits local linearizations and also thatM[H, G] depends continuously on the pair (H, G). Moreover, EMFCQ is shown to be satisfied generically.The authors would like to thank Rainer Hettich and Doug Ward for fruitful discussions. Moreover, the authors are indebted to the anonymous referees for their valuable comments.