Convergence of an Interior Point Algorithm for Continuous Minimax
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We propose an algorithm for the constrained continuous minimax problem. The algorithm uses a quasi-Newton search direction, based on subgradient information, conditional on maximizers. The initial problem is transformed to an equivalent equality constrained problem, where the logarithmic barrier function is used to ensure feasibility. In the case of multiple maximizers, the algorithm adopts semi-infinite programming iterations toward epiconvergence. Satisfaction of the equality constraints is ensured by an adaptive quadratic penalty function. The algorithm is augmented by a discrete minimax procedure to compute the semi-infinite programming steps and ensure overall progress when required by the adaptive penalty procedure. Progress toward the solution is maintained using merit functions.
KeywordsWorst case analysis Continuous minimax algorithms Interior point methods Semi–infinite programming
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- 1.Rustem, B., Zakovic, S., Parpas, P.: An interior point algorithm for continuous minimax: implementation and computation. Comput. Optim. Appl. (2008, to appear) Google Scholar
- 3.Polak, E., Mayne, D.Q., Higgins, E.J.: A superlinearly convergent minimax algorithm for minimax problems. UCB/ERL M86/103, Department of Electrical Engineering, University California, Berkeley, CA (1988) Google Scholar
- 5.Obasanjo, E., Rustem, B.: An interior point algorithm for nonlinear minimax problems (to appear) Google Scholar
- 6.Demyanov, V.F., Pvnyi, A.B.: Numerical methods for finding saddle points. USSR Comput. Math. Math. Phys. 12, 1099–1127 (1972) Google Scholar
- 7.Rustem, B., Howe, M.A.: Algorithms for Worst-case Design with Applications to Risk Management. Princeton University Press, Princeton (2001) Google Scholar
- 15.Powell, M.J.D.: A fast algorithm for nonlinearly constrained optimization calculations. In: Lecture Notes in Mathematics, vol. 630, pp. 144–157. Springer, Berlin (1978) Google Scholar
- 17.Parpas, P., Rustem, B.: An algorithm for the global optimization of a class of continuous minimax problems. J. Optim. Theory Appl. (2008, to appear) Google Scholar
- 19.Yamashita, H.: A globally convergent primal-dual interior point method for constrained optimization. Technical Report, Mathematical Systems Institute Inc, 2-5-3 Shinjuku, Shinjuku-ku, Tokyo, Japan (May 1995) Google Scholar