A barrier function method for minimax problems

This paper presents an algorithm based on barrier functions for solving semi-infinite minimax problems which arise in an engineering design setting. The algorithm bears a resemblance to some of the current interior penalty function methods used to solve constrained minimization problems. Global convergence is proven, and numerical results are reported which show that the algorithm is exceptionally robust, and that its performance is comparable, while its structure is simpler than that of current first-order minimax algorithms.

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

[2]  K. Madsen.  An Algorithm for Minimax Solution of Overdetermined Systems of Non-linear Equations , 1975 .

[3]  Elijah Polak,et al.  A method of feasible directions using function approximations, with applications to min max problems☆ , 1973 .

[4]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[5]  S. P. Han,et al.  Variable metric methods for minimizing a class of nondifferentiable functions , 1977, Math. Program..

[6]  E. Polak,et al.  A recursive quadratic programming algorithm for semi-infinite optimization problems , 1982 .

[7]  D. Mayne,et al.  Adaptive control of ARMA plants using worst-case design by semi-infinite optimization , 1987 .

[8]  A. Conn,et al.  An Efficient Method to Solve the Minimax Problem Directly , 1978 .

[9]  R. Fletcher,et al.  An algorithm for composite nonsmooth optimization problems , 1986 .

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

[11]  Anthony V. Fiacco,et al.  The Sequential Unconstrained Minimization Technique (SUMT) Without Parameters , 1967, Oper. Res..

[12]  Sanjay Mehrotra,et al.  Relaxed variants of Karmarkar's algorithm for linear programs with unknown optimal objective value , 1988, Math. Program..

[13]  Nicholas I. M. Gould,et al.  An exact penalty function for semi-infinite programming , 1987, Math. Program..

[14]  L. Armijo.  Minimization of functions having Lipschitz continuous first partial derivatives. , 1966 .

[15]  E. Polak,et al.  A quadratically convergent algorithm for solving infinite dimensional inequalities , 1982 .

[16]  Elijah Polak,et al.  On the design of stabilizing compensators via semi-infinite optimization , 1989 .

[17]  R. Mifflin.  Rates of convergence for a method of centers algorithm , 1976 .

[18]  R. Hettich,et al.  On quadratically convergent methods for semi-infinite programming , 1979 .

[19]  G. Sonnevend.  New Algorithms in Convex Programming Based on a Notion of “Centre” (for Systems of Analytic Inequalities) and on Rational Extrapolation , 1988 .

[20]  J. Stoer,et al.  An implementation of the method of analytic centers , 1988 .

[21]  D. Mayne,et al.  An algorithm for optimization problems with functional inequality constraints , 1976 .

[22]  C. C. Gonzaga,et al.  An Algorithm for Solving Linear Programming Problems in O(n 3 L) Operations , 1989 .

[23]  Kaj Madsen,et al.  Combined lp and quasi-Newton methods for minimax optimization , 1981, Math. Program..

[24]  J. Stoer,et al.  Global ellipsoidal approximations and homotopy methods for solving convex analytic programs , 1990 .

[25]  B. N. Pshenichnyi,et al.  Numerical Methods in Extremal Problems. , 1978 .

[26]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[27]  G. Alistair Watson,et al.  A projected lagrangian algorithm for semi-infinite programming , 1985, Math. Program..

[28]  Åke Björck,et al.  Numerical Methods , 2021, Markov Renewal and Piecewise Deterministic Processes.

[29]  M. R. Osborne,et al.  Trajectory analysis and extrapolation in barrier function methods , 1978, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[30]  M. Fukushima,et al.  A comparative study of several semi-infinite nonlinear programming algorithms , 1988 .

[31]  W. Murray,et al.  A Projected Lagrangian Algorithm for Nonlinear Minimax Optimization , 1980 .