Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques

We develop two implementable algorithms, the first for the solution of finite and the second for the solution of semi-infinite min-max-min problems. A smoothing technique (together with discretization for the semi-infinite case) is used to construct a sequence of approximating finite min-max problems, which are solved with increasing precision. The smoothing and discretization approximations are initially coarse, but are made progressively finer as the number of iterations is increased. This reduces the potential ill-conditioning due to high smoothing precision parameter values and computational cost due to high levels of discretization. The behavior of the algorithms is illustrated with three semi-infinite numerical examples.

[1]  L. Shepp,et al.  OPTIMAL PATHS FOR A CAR THAT GOES BOTH FORWARDS AND BACKWARDS , 1990 .

[2]  Elijah Polak,et al.  Optimization: Algorithms and Consistent Approximations , 1997 .

[3]  Algorithms for optimization problems with exclusion constraints , 1986 .

[4]  Chung-Kuan Cheng,et al.  Symbolic layout compaction under conditional design rules , 1992, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[5]  B. V. K. Vijaya Kumar,et al.  Path Planning for Autonomous Robots Using Neural Networks , 1997 .

[6]  Elijah Polak.  Smoothing Techniques for the Solution of Finite and Semi-Infinite Min-Max-Min Problems , 2003 .

[7]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[8]  Johannes O. Royset,et al.  Reliability-based optimal structural design by the decoupling approach , 2001, Reliab. Eng. Syst. Saf..

[9]  E. Polak,et al.  Theoretical and computational aspects of the optimal design centering, tolerancing, and tuning problem , 1979 .

[10]  Li Xingsi,et al.  AN ENTROPY-BASED AGGREGATE METHOD FOR MINIMAX OPTIMIZATION , 1992 .

[11]  Adil M. Bagirov.  Minimization Methods for One Class of Nonsmooth Functions and Calculation of Semi-Equilibrium Prices , 1999 .

[12]  On the optimal design centering, tolerancing, and tuning problem , 1985 .

[13]  Elijah Polak,et al.  On the Conversion of Optimization Problems with Max-Min Constraints to Standard Optimization Problems , 1998, SIAM J. Optim..

[14]  John W. Bandler,et al.  A nonlinear programming approach to optimal design centering, tolerancing, and tuning , 1976 .

[15]  Hans Bock,et al.  Path Planning For Satellite Mounted Robots , 1970 .

[16]  A. Voreadis,et al.  A cut-map algorithm for design problems with parameter tolerances and tuning , 1982 .

[17]  S. Hakimi,et al.  Globally optimal floorplanning for a layout problem , 1996 .

[18]  Michael K. H. Fan,et al.  On convex formulation of the floorplan area minimization problem , 1998, ISPD '98.