Semi-infinite Optimization Problem Research Articles

Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems

In this paper we derive a priori error estimates for linear-quadratic elliptic optimal control problems with finite dimensional control space and state constraints in the whole domain, which can be written as semi-infinite optimization problems. Numerical experiments are conducted to underline our theory.

A Nonsmooth Newton Method with Path Search and Its Use in Solving $C^{1,1}$ Programs and Semi-Infinite Problems

In [S. Butikofer, Math. Methods Oper. Res., 68 (2008), pp. 235-256] a nonsmooth Newton method globalized with the aid of a path search was developed in an abstract framework. We refine the convergence analysis given there and adapt this algorithm to certain finite dimensional optimization problems with $C^{1,1}$ data. Such problems arise, for example, in semi-infinite programming under a reduction approach without strict complementarity and in generalized Nash equilibrium models. Using results from parametric optimization and variational analysis, we work out in detail the concrete Newton schemes and the construction of a path for these applications and discuss a series of numerical results for semi-infinite and generalized semi-infinite optimization problems.

Robust stochastic dominance and its application to risk-averse optimization

We introduce a new preference relation in the space of random variables, which we call robust stochastic dominance. We consider stochastic optimization problems where risk-aversion is expressed by a robust stochastic dominance constraint. These are composite semi-infinite optimization problems with constraints on compositions of measures of risk and utility functions. We develop necessary and sufficient conditions of optimality for such optimization problems in the convex case. In the nonconvex case, we derive necessary conditions of optimality under additional smoothness assumptions of some mappings involved in the problem.

Optimal switching instants for a switched-capacitor DC/DC power converter

We consider a switched-capacitor DC/DC power converter with variable switching instants. The determination of optimal switching instants giving low output ripple and strong load regulation is posed as a non-smooth dynamic optimization problem. By introducing a set of auxiliary differential equations and applying a time-scaling transformation, we formulate an equivalent optimization problem with semi-infinite constraints. Existing algorithms can be applied to solve this smooth semi-infinite optimization problem. The existence of an optimal solution is also established. For illustration, the optimal switching instants for a practical switched-capacitor DC/DC power converter are determined using this approach.

Global Optimum Design of Uniform FIR Filter Bank With Magnitude Constraints

The optimum design of a uniform finite impulse response filter bank can be formulated as a nonlinear semi-infinite optimization problem. However, this optimization problem is nonconvex with infinitely many inequality constraints. In this paper, we propose a new hybrid approach for solving this highly challenging nonlinear, nonconvex semi-infinite optimization problem. In this approach, a gradient-based method is used in conjunction with a filled function method to determine a global minimum of the problem. This new hybrid approach finds an optimal result independent of the initial guess of the solution. The method is applied to some existing examples. The results obtained are superior to those obtained by other existing methods.

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.

Global weak sharp minima for convex (semi-)infinite optimization problems

We mainly consider global weak sharp minima for convex infinite and semi-infinite optimization problems (CIP). In terms of the normal cone, subdifferential and directional derivative, we provide several characterizations for (CIP) to have global weak sharp minimum property.

Lipschitz modulus in convex semi-infinite optimizationviad.c. functions

We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convex semi-infinite optimization problems. Specifically, the paper provides a lower and an upper bound for this modulus, both of them given exclusively in terms of the problem's data. Moreover, the upper bound is shown to be the exact modulus when the number of constraints is finite. In the particular case of linear problems the upper bound (or exact modulus) adopts a notably simplified expression. Our approach is based on variational techniques applied to certain difference of convex functions related to the model. Some results of (M.J. Canovas et al., J. Optim. Theory Appl. (2008) Online First) (which go back to (M.J. Canovas, J. Global Optim. 41 (2008) 1-13) and (Ioffe, Math. Surveys 55 (2000) 501-558; Control Cybern. 32 (2003) 543-554)) constitute the starting point of the present work.

Stability of Indices in the KKT Conditions and Metric Regularity in Convex Semi-Infinite Optimization

This paper deals with a parametric family of convex semi-infinite optimization problems for which linear perturbations of the objective function and continuous perturbations of the right-hand side of the constraint system are allowed. In this context, Canovas et al. (SIAM J. Optim. 18:717–732, [2007]) introduced a sufficient condition (called ENC in the present paper) for the strong Lipschitz stability of the optimal set mapping. Now, we show that ENC also entails high stability for the minimal subsets of indices involved in the KKT conditions, yielding a nice behavior not only for the optimal set mapping, but also for its inverse. Roughly speaking, points near optimal solutions are optimal for proximal parameters. In particular, this fact leads us to a remarkable simplification of a certain expression for the (metric) regularity modulus given in Canovas et al. (J. Glob. Optim. 41:1–13, [2008]) (and based on Ioffe (Usp. Mat. Nauk 55(3):103–162, [2000]; Control Cybern. 32:543–554, [2003])), which provides a key step in further research oriented to find more computable expressions of this regularity modulus.

First-order optimality conditions for two classes of generalized nonsmooth semi-infinite optimization

We investigate two classes of generalized nonsmooth semi-infinite optimization problems in this paper, that is, the generalized convex semi-infinite optimization problem and the generalized Lipscitz semi-infinite optimization problem. Their first order necessary optimality conditions are obtained using either the differentiability properties of the optimal value functions or the bounds for the directional derivatives of the optimal value function.

Complexity aspects of a semi-infinite optimization problem†

This article surveys some of the author's work relating real number complexity and optimization theory. The focus is on the interplay between structural complexity considerations and semi-infinite optimization. †Dedicated to H.Th. Jongen on the occasion of his 60th birthday

The Adaptive Convexification Algorithm: A Feasible Point Method for Semi-Infinite Programming

We present a new numerical solution method for semi-infinite optimization problems. Its main idea is to adaptively construct convex relaxations of the lower level problem, replace the relaxed lower level problems equivalently by their Karush–Kuhn–Tucker conditions, and solve the resulting mathematical programs with complementarity constraints. This approximation produces feasible iterates for the original problem. The convex relaxations are constructed with ideas from the $\alpha$BB method of global optimization. The necessary upper bounds for second derivatives of functions on box domains can be determined using the techniques of interval arithmetic, where our algorithm already works if only one such bound is available for the problem. We show convergence of stationary points of the approximating problems to a stationary point of the original semi-infinite problem within arbitrarily given tolerances. Numerical examples from Chebyshev approximation and design centering illustrate the performance of the method.

Fuzzy matrix games via a fuzzy relation approach

A generalized model for a two person zero sum matrix game with fuzzy goals and fuzzy payoffs via fuzzy relation approach is introduced, and it is shown to be equivalent to two semi-infinite optimization problems. Further, in certain special cases, it is observed that the two semi-infinite optimization problems reduce to (finite) linear programming problems which are dual to each other either in the fuzzy sense or in the crisp sense.

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: 10.1007/s10898-007-9232-3 ), 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.

Smoothing by mollifiers. Part I: semi-infinite optimization

We show 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. This function is constructed as the mollification of the lower level optimal value function. Moreover, we use correspondences between Karush---Kuhn---Tucker points of the original and the smoothed problem, and between their associated Morse indices, to prove the connectedness of the so-called min---max digraph for semi-infinite problems.

Lipschitz behavior of convex semi-infinite optimization problems: a variational approach

In this paper we make use of subdifferential calculus and other variational techniques, traced out from [Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 3(333), 103---162; Engligh translation Math. Surveys 55, 501---558 (2000); Ioffe, A.D.: On rubustness of the regularity property of maps. Control cybernet 32, 543---554 (2003)], to derive different expressions for the Lipschitz modulus of the optimal set mapping of canonically perturbed convex semi-infinite optimization problems. In order to apply this background for obtaining the modulus of metric regularity of the associated inverse multifunction, we have to analyze the stable behavior of this inverse mapping. In our semi-infinite framework this analysis entails some specific technical difficulties. We also provide a new expression of a global variational nature for the referred regularity modulus.

Lagrange Multipliers in Nonsmooth Semi-Infinite Optimization Problems

Using the variational analysis technique, in terms of the epi-coderivative, we provide Lagrange multiplier rules for a class of semi-infinite optimization problems where all functions are lower semicontinuous or locally Lipschitz.

Weak Sharp Minima for Semi-infinite Optimization Problems with Applications

We study local weak sharp minima and sharp minima for smooth semi-infinite optimization problems SIP. We provide several dual and primal characterizations for a point to be a sharp minimum or a weak sharp minimum of SIP. As applications, we present several sufficient and necessary conditions of calmness for infinitely many smooth inequalities. In particular, we improve some calmness results in [R. Henrion and J. Outrata, Math. Program., 104 (2005), pp. 437–464].

Metric Regularity in Convex Semi-Infinite Optimization under Canonical Perturbations

This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right-hand side of the constraints and linear perturbations of the objective function. In this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the Slater constraint qualification, together with a certain additional requirement in the Karush-Kuhn-Tucker conditions. For linear problems this sufficient condition turns out to be also necessary for the metric regularity, and it is equivalent to some well-known stability concepts.

Optimal actuator/sensor placement for linear parabolic PDEs using spatial [formula omitted] norm

The present work focuses on the optimal, with respect to certain criteria, placement of control actuators and sensors for transport-reaction processes which are mathematically modeled by linear parabolic partial differential equations. Using modal decomposition to discretize the spatial coordinate, and the notions of spatial and modal controllability and observability, the semi-infinite optimization problem is formulated as a nonlinear optimization problem in an abstract space, which is subsequently solved using standard search algorithms. The proposed method is successfully applied to a representative process, modeled by a one-dimensional parabolic PDE, where the optimal location of a single and multiple point actuators are computed.