A Merit Function Approach for Direct Search

In this paper it is proposed to equip direct-search methods with a general procedure to minimize an objective function, possibly nonsmooth, without using derivatives and subject to constraints on the variables. One aims at considering constraints, most likely nonlinear or nonsmooth, for which the derivatives of the corresponding functions are also unavailable. The novelty of this contribution relies mostly on how relaxable constraints are handled. Such constraints, which can be relaxed during the course of the optimization, are taken care of by a merit function and, if necessary, by a restoration procedure. Constraints that are unrelaxable, when present, are treated by an extreme barrier approach. One is able to show that the resulting merit function direct-search algorithm exhibits global convergence properties for first-order stationary constraints. As in the progressive barrier method [C. Audet and J. E. Dennis Jr., SIAM J. Optim., 20 (2009), pp. 445--472], we provide a mechanism to indicate the transf...

[1]  John L. Nazareth,et al.  Introduction to derivative-free optimization , 2010, Math. Comput..

[2]  Charles Audet,et al.  Analysis of Generalized Pattern Searches , 2000, SIAM J. Optim..

[3]  Paul Tseng,et al.  Objective-derivative-free methods for constrained optimization , 2002, Math. Program..

[4]  M. Powell.  A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation , 1994 .

[5]  Charles Audet,et al.  A Pattern Search Filter Method for Nonlinear Programming without Derivatives , 2001, SIAM J. Optim..

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

[7]  Benoît Colson,et al.  Trust-region algorithms for derivative-free optimization and nonlinear bilevel programming , 2004, 4OR.

[8]  J. M. Martínez,et al.  Derivative-free methods for nonlinear programming with general lower-level constraints , 2011 .

[9]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[10]  J. Jahn.  Introduction to the Theory of Nonlinear Optimization , 1994 .

[11]  Charles Audet,et al.  Globalization strategies for Mesh Adaptive Direct Search , 2008, Comput. Optim. Appl..

[12]  Hugues Bersini,et al.  Constrained, non-linear, derivative-free, parallel optimization of continuous, high computing load, noisy objective functions , 2004 .

[13]  J. Dennis,et al.  Direct Search Methods for Nonlinearly Constrained Optimization Using Filters and Frames , 2004 .

[14]  Charles Audet,et al.  Nonsmooth optimization through Mesh Adaptive Direct Search and Variable Neighborhood Search , 2006, J. Glob. Optim..

[15]  Katya Scheinberg,et al.  A derivative free optimization algorithm in practice , 1998 .

[16]  Herbert Hamers,et al.  Constrained optimization involving expensive function evaluations: A sequential approach , 2005, Eur. J. Oper. Res..

[17]  Stefano Lucidi,et al.  A Derivative-Free Algorithm for Inequality Constrained Nonlinear Programming via Smoothing of an linfty Penalty Function , 2009, SIAM J. Optim..

[18]  Charles Audet,et al.  Mesh Adaptive Direct Search Algorithms for Constrained Optimization , 2006, SIAM J. Optim..

[19]  José Mario Martínez,et al.  Constrained derivative-free optimization on thin domains , 2012, Journal of Global Optimization.

[20]  Robert Michael Lewis,et al.  A Globally Convergent Augmented Lagrangian Pattern Search Algorithm for Optimization with General Constraints and Simple Bounds , 2002, SIAM J. Optim..

[21]  T. Kolda,et al.  A generating set direct search augmented Lagrangian algorithm for optimization with a combination of general and linear constraints , 2006 .

[22]  Charles Audet,et al.  A MADS Algorithm with a Progressive Barrier for Derivative-Free Nonlinear Programming , 2007 .

[23]  O. SIAMJ..  A DERIVATIVE-FREE ALGORITHM FOR INEQUALITY CONSTRAINED NONLINEAR PROGRAMMING VIA SMOOTHING OF AN ∞ PENALTY FUNCTION∗ , 2009 .

[24]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[25]  Marco Sciandrone,et al.  Sequential Penalty Derivative-Free Methods for Nonlinear Constrained Optimization , 2010, SIAM J. Optim..

[26]  Luís N. Vicente,et al.  Analysis of direct searches for discontinuous functions , 2012, Math. Program..