On smooth reformulations and direct non-smooth computations for minimax problems

Minimax problems can be approached by reformulating them into smooth problems with constraints or by dealing with the non-smooth objective directly. We focus on verified enclosures of all globally optimal points of such problems. In smooth problems in branch and bound algorithms, interval Newton methods can be used to verify existence and uniqueness of solutions, to be used in eliminating regions containing such solutions, and point Newton methods can be used to obtain approximate solutions for good upper bounds on the global optimum. We analyze smooth reformulation approaches, show weaknesses in them, and compare reformulation to solving the non-smooth problem directly. In addition to analysis and illustrative problems, we exhibit the results of numerical computations on various test problems.

[1]  Ralph Baker Kearfott,et al.  On rigorous upper bounds to a global optimum , 2014, J. Glob. Optim..

[2]  R. Baker Kearfott,et al.  The cluster problem in multivariate global optimization , 1994, J. Glob. Optim..

[3]  R. Baker Kearfott,et al.  Validated Linear Relaxations and Preprocessing: Some Experiments , 2005, SIAM J. Optim..

[4]  G. A. Watson,et al.  Approximation in normal linear spaces , 2000 .

[5]  Ralph Baker Kearfott,et al.  Interval Computations, Rigor and Non-Rigor in Deterministic Continuous Global Optimization , 2010 .

[6]  R. Baker Kearfott,et al.  Slope Intervals, Generalized Gradients, Semigradients, Slant Derivatives, and Csets , 2004, Reliab. Comput..

[7]  Eldon Hansen,et al.  Global optimization using interval analysis , 1992, Pure and applied mathematics.

[8]  A. Neumaier,et al.  Solving minimax problems by interval methods , 1990 .

[9]  R. B. Kearfott.  Rigorous Global Search: Continuous Problems , 1996 .

[10]  Jorge J. Moré,et al.  User guide for MINPACK-1. [In FORTRAN] , 1980 .

[11]  G. Watson.  The Minimax Solution of an Overdetermined System of Non-linear Equations , 1979 .

[12]  R. Baker Kearfott,et al.  Discussion and empirical comparisons of linear relaxations and alternate techniques in validated deterministic global optimization , 2006, Optim. Methods Softw..

[13]  C. Lemaréchal.  Nondifferentiable optimization , 1989 .

[14]  R. Baker Kearfott,et al.  GlobSol user guide , 2009, Optim. Methods Softw..

[15]  Luc Jaulin.  Reliable Minimax Parameter Estimation , 2001, Reliab. Comput..

[16]  R. B. Kearfott.  SLOPE INTERVAL, GENERALIZED GRADIENT, SEMIGRADIENT, AND SLANT DERIVATIVE , 2001 .

[17]  R. B. Kearfott,et al.  Optimal preconditioners for the interval newton method , 1990 .

[18]  Jorge J. Moré,et al.  User Guide for Minpack-1 , 1980 .

[19]  Baker Kearfott.  A proof of convergence and an error bound for the method of bisection in ⁿ , 1978 .

[20]  A. Neumaier.  Interval methods for systems of equations , 1990 .