Steering Exact Penalty Methods for Optimization

This paper reviews, extends and analyzes a new class of penalty methods for nonlinear optimization. These methods adjust the penalty parameter dynamically; by controlling the degree of linear feasibility achieved at every iteration, they promote balanced progress toward optimality and feasibility. In contrast with classical approaches, the choice of the penalty parameter ceases to be a heuristic and is determined, instead, by a subproblem with clearly defined objectives. The new penalty update strategy is presented in the context of sequential quadratic programming (SQP) and sequential linear-quadratic programming (SLQP) methods that use trust regions to promote convergence. The paper concludes with a discussion of penalty parameters for merit functions used in line search methods. Department of Computer Science, University of Colorado, Boulder, CO 80309. This author was supported by Army Research Office Grants DAAD19-02-1-0407, and by National Science Foundation grants CCR-0219190 and CHE-0205170. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, 60208-3118, USA. These authors were supported by National Science Foundation grant CCR-0219438 and Department of Energy grant DE-FG02-87ER25047-A004.

[1]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[2]  A. Sartenaer,et al.  Automatic decrease of the penalty parameter in exact penalty function methods , 1995 .

[3]  D. Ralph,et al.  Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints , 2004 .

[4]  Roger Fletcher,et al.  Nonlinear programming and nonsmooth optimization by successive linear programming , 1989, Math. Program..

[5]  Jorge Nocedal,et al.  On the Implementation of an Algorithm for Large-Scale Equality Constrained Optimization , 1998, SIAM J. Optim..

[6]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[7]  M. Anitescu.  On Solving Mathematical Programs With Complementarity Constraints As Nonlinear Programs , 2002 .

[8]  Jorge Nocedal,et al.  Interior Methods for Mathematical Programs with Complementarity Constraints , 2006, SIAM J. Optim..

[9]  M. El-Alem.  A global convergence theory for the Celis-Dennis-Tapia trust-region algorithm for constrained optimization , 1991 .

[10]  E. Omojokun.  Trust region algorithms for optimization with nonlinear equality and inequality constraints , 1990 .

[11]  Mihai Anitescu,et al.  Global Convergence of an Elastic Mode Approach for a Class of Mathematical Programs with Complementarity Constraints , 2005, SIAM J. Optim..

[12]  Roger Fletcher,et al.  On the global convergence of an SLP–filter algorithm that takes EQP steps , 2003, Math. Program..

[13]  Donald Goldfarb,et al.  Interior-point ℓ2-penalty methods for nonlinear programming with strong global convergence properties , 2006, Math. Program..

[14]  Sven Leyffer,et al.  Nonlinear programming without a penalty function , 2002, Math. Program..

[15]  Jorge Nocedal,et al.  An interior algorithm for nonlinear optimization that combines line search and trust region steps , 2006, Math. Program..

[16]  Ya-Xiang Yuan,et al.  A trust region algorithm for equality constrained optimization , 1990, Math. Program..

[17]  Arne Drud,et al.  CONOPT: A GRG code for large sparse dynamic nonlinear optimization problems , 1985, Math. Program..

[18]  Jimmie D. Lawson,et al.  Presentation , 2000, MFCSIT.

[19]  Jorge Nocedal,et al.  An Interior Point Algorithm for Large-Scale Nonlinear Programming , 1999, SIAM J. Optim..

[20]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[21]  James V. Burke,et al.  A Robust Trust Region Method for Constrained Nonlinear Programming Problems , 1992, SIAM J. Optim..

[22]  Richard A. Tapia,et al.  A trust region strategy for nonlinear equality constrained op-timization , 1984 .

[23]  Robert J. Vanderbei,et al.  Interior-Point Algorithms, Penalty Methods and Equilibrium Problems , 2006, Comput. Optim. Appl..

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

[25]  Stefan Scholtes,et al.  Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity , 2000, Math. Oper. Res..

[26]  Nicholas I. M. Gould,et al.  An algorithm for nonlinear optimization using linear programming and equality constrained subproblems , 2004, Math. Program..

[27]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[28]  Nicholas I. M. Gould,et al.  On the Convergence of Successive Linear-Quadratic Programming Algorithms , 2005, SIAM J. Optim..

[29]  Jon W. Tolle,et al.  Exact penalty functions in nonlinear programming , 1973, Math. Program..

[30]  John E. Dennis,et al.  A Global Convergence Theory for General Trust-Region-Based Algorithms for Equality Constrained Optimization , 1997, SIAM J. Optim..