A dual parameterization approach to linear-quadratic semi-infinite programming problems

Semi-infinite programming problems are special optimization problems in which a cost is to be minimized subject to infinitely many constraints. This class of problems has many real-world applications. In this paper, we consider a class of linear-quadratic semi-infinite programming problems. Using the duality theory, the dual problem is obtained, where the decision variables are measures. A new parameterization scheme is developed for approximating these measures. On this bases, an efficient algorithm for computing the solution of the dual problem is obtained. Rigorous convergence results are given to support the algorithm. The solution of the primal problem is easily obtained from that of the dual problem. For illustration, three numerical examples are included.

[1]  E. Polak.  On the mathematical foundations of nondifferentiable optimization in engineering design , 1987 .

[2]  Thomas F. Coleman,et al.  Optimization Toolbox User's Guide , 1998 .

[3]  Kok Lay Teo,et al.  A computational algorithm for functional inequality constrained optimization problems , 1990, Autom..

[4]  A. Tits,et al.  A globally convergent algorithm with adaptively refined discretization for semi-infinite optimization problems arising in engineering design , 1989 .

[5]  E. Polak,et al.  Control system design via semi-infinite optimization: A review , 1984, Proceedings of the IEEE.

[6]  P. R. Gribik,et al.  A central-cutting-plane algorithm for semi-infinite programming problems , 1979 .

[7]  Kok Lay Teo,et al.  Continuous-time envelope constrained filter design via orthonormal filters , 1995 .

[8]  R. A. Silverman,et al.  Introductory Real Analysis , 1972 .

[9]  Dual Quasi-Newton Algorithm for Infinitely Constrained Optimization Problems , 1991 .

[10]  Kenneth O. Kortanek,et al.  A Central Cutting Plane Algorithm for Convex Semi-Infinite Programming Problems , 1993, SIAM J. Optim..

[11]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

[12]  J. E. Falk,et al.  Infinitely constrained optimization problems , 1976 .

[13]  Kenneth O. Kortanek,et al.  Semi-Infinite Programming: Theory, Methods, and Applications , 1993, SIAM Rev..

[14]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[15]  Kok Lay Teo,et al.  A Unified Computational Approach to Optimal Control Problems , 1991 .

[16]  Kenneth O. Kortanek,et al.  Numerical treatment of a class of semi‐infinite programming problems , 1973 .

[17]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[18]  C. C. Gonzaga,et al.  An improved algorithm for optimization problems with functional inequality constraints , 1980 .