Constrained Minimax Problems Research Articles

Generalized monotone line search SQP algorithm for constrained minimax problems

In this article, non-linear minimax problems with general constraints are discussed. By means of solving one quadratic programming an improved direction is yielded and a second-order correction direction can also be at hand via one system of linear equations. So a new algorithm for solving the discussed problems is presented. In connection with a special merit function, the generalized monotone line search is used to yield the step size at each iteration. Under mild conditions, we can ensure global and superlinear convergence. Finally, some numerical experiments are operated to test our algorithm, and the results demonstrate that it is promising.

Continuous minimax optimization using modal intervals

Many real life problems can be stated as a continuous minimax optimization problem. Well-known applications to engineering, finance, optics and other fields demonstrate the importance of having reliable methods to tackle continuous minimax problems. In this paper a new approach to the solution of continuous minimax problems over reals is introduced, using tools based on modal intervals. Continuous minimax problems, and global optimization as a particular case, are stated as the computation of semantic extensions of continuous functions, one of the key concepts of modal intervals. Modal intervals techniques allow to compute, in a guaranteed way, such semantic extensions by means of an efficient algorithm. Several examples illustrate the behavior of the algorithms in unconstrained and constrained minimax problems.

Nonmonotone Line Search Algorithm for Constrained Minimax Problems

In this paper, an algorithm for constrained minimax problems is presented which is globally convergent and whose rate of convergence is two-step superlinear. The algorithm applies SQP to the constrained minimax problems by combining a nonmonotone line search and a second-order correction technique, which guarantees a full steplength while close to a solution, such that the Maratos effect is avoided and two-step superlinear convergence is achieved.

A Sharp Lagrange Multiplier Rule for Nonsmooth Mathematical Programming Problems Involving Equality Constraints

It is shown that a Lagrange multiplier rule that uses approximate Jacobians holds for mathematical programming problems involving Lipschitzian functions, finitely many equality constraints, and convex set constraints. It is sharper than the corresponding Lagrange multiplier rules for the convex-valued subdifferentials such as those of Clarke [Optimization and Nonsmooth Analysis, 2nd ed., SIAM, 1990] and Michel and Penot [Differential Integral Equations, 5 (1992), pp. 433--454]. The Lagrange multiplier result is obtained by means of a controllability criterion and the theory of fans developed by A. D. Ioffe [Math. Oper. Res., 9 (1984), pp. 159--189, Math. Programming, 58 (1993), pp. 137--145]. As an application, necessary optimality conditions are derived for a class of constrained minimax problems. An example is discussed to illustrate the nature of the multiplier rule.

An ODE-based approach to nonlinearly constrained minimax problems

We consider the following problem: Choosex1, ...,xn to Open image in new window wherem1,m2,m3 are integers with 0≤m1≤m2≤m3, thefi are given real numbers, and thegi are given smooth functions. Constraints of the formgi(x1, ...,xn)=0 can also be handled without problem. Each iteration of our algorithm involves approximately solving a certain non-linear system of first-order ordinary differential equations to get a search direction for a line search and using a Newton-like approach to correct back into the feasible region when necessary. The algorithm and our Fortran implementation of it will be discussed along with some examples. Our experience to date has been that the program is more robust than any of the library routines we have tried, although it generally requires more computer time. We have found this program to be an extremely useful tool in diverse areas, including polymer rheology, computer vision, and computation of convexity-preserving rational splines.

Solving large-scale minimax problems with the primal—dual steepest descent algorithm

This paper shows that the primal-dual steepest descent algorithm developed by Zhu and Rockafellar for large-scale extended linear--quadratic programming can be used in solving constrained minimax problems related to a generalC2 saddle function. It is proved that the algorithm converges linearly from the very beginning of the iteration if the related saddle function is strongly convex--concave uniformly and the cross elements between the convex part and the concave part of the variables in its Hessian are bounded on the feasible region. Better bounds for the asymptotic rates of convergence are also obtained. The minimax problems where the saddle function has linear cross terms between the convex part and the concave part of the variables are discussed specifically as a generalization of the extended linear--quadratic programming. Some fundamental features of these problems are laid out and analyzed.

Avoiding the Maratos Effect by Means of a Nonmonotone Line Search. II. Inequality Constrained Problems—Feasible Iterates

When solving inequality constrained optimization problems via Sequential Quadratic Programming (SQP), it is potentially advantageous to generate iterates that all satisfy the constraints: all quadratic programs encountered are then feasible and there is no need for a surrogate merit function. (Feasibility of the successive iterates is in fact required in many contexts such as in real-time applications or when the objective function is not defined outside the feasible set.) It has recently been shown that this is, indeed, possible, by means of a suitable perturbation of the original SQP iteration, without losing superlinear convergence. In this context, the well-known Maratos effect is compounded by the possible infeasibility of the full step of one even close to a solution. These difficulties have been accommodated by making use of a suitable modification of a “bending” technique proposed by Mayne and Polak, requiring evaluation of the constraints function at an auxiliary point at each iteration. In Part I of this two-part paper, it was shown that, when feasibility of the successive iterates is not required, the Maratos effect can be avoided by combining Mayne and Polak’s technique with a non-monotone line search proposed by Grippo, Lampariello, and Lucidi in the context of unconstrained optimization, in such a way that, except possibly at a few early iterations, function evaluations are no longer performed at auxiliary points. In this second part, it is shown that feasibility can be restored without resorting to additional constraint evaluations, by adaptively estimating a bound on the second derivatives of the active constraints. Extension to constrained minimax problems is briefly discussed.

An algorithm for a constrained saddle point problem via an exact penalty method

In this paper we present an algorithm which allows the approximation of a constrained saddle point problem by a sequence of unconstrained and finite dimensional optimization problems using an exact penalty functional for constrained minimax problems. Numerical experiments, both in finite and infinite dimension, are reported and are compared with those obtained by the use of an external penalization.

On penalty methods for minimax problems

Penalty algorithms for constrained minimax problems typically involve a sequence of unconstrained approximates which is pointwise monotone in each variable. This paper generalizes convergence results for a wider class of algorithms while imposing conditions which are close to being minimal.

An algorithm for solving linearly constrained minimax problems

In this study, we propose an algorithm for solving a minimax problem over a polyhedral set defined in terms of a system of linear inequalities. At each iteration a direction is found by solving a quadratic programming problem and then a suitable step size along that direction is taken through an extension of Armijo's approximate line search technique. We show that each accumulation point is a Kuhn-Tucker solution and give a condition that guarantees convergence of the whole sequence of iterations. Through the use of an exact penalty function, the algorithm can be used for solving constrained nonlinear programming. In this case, our algorithm resembles that of Han, but differs from it both in the direction-finding and the line search steps.

Practical application of a penalty function approach to constrained minimax optimization

A practical, penalty function approach to solving constrained minimax problems is applied here. In essence, this approach reformulates the constrained minimax problem as an unconstrained minimax problem. A recently proposed optimization algorithm called grazor search is used to solve the reformulated unconstrained minimax problem. The proposed approach can handle inequality constraints-parameter constraints in particular. A practical transmission-line filter example with parameter constraints illustrates the results.

A Unified Review of Optimization

The main objective of this paper is to give a survey of recent automatic optimization methods which either have found or should find useful application in the area of computer-aided network design. Huang's family of algorithms for unconstrained optimization is reviewed. The Fletcher method and the Charalambous family of algorithms for unconstrained optimization, which abandon the "full linear search," are presented. Special emphasis is devoted to algorithms by Bandler and Charalambous on least pth and minimax optimization which can be readily programmed and used. Due to work by Bandler and Charalambous, it is shown how constrained minimax problems can be solved exactly as unconstrained minimax problems by using a new approach to nonlinear programming. The application of minirnax optimization on the design of lumped-distributed active filters, problems for future investigation, and a select list of references are also included.