Partially Strictly Monotone and Nonlinear Penalty Functions for Constrained Mathematical Programs

  • Authors:

  • X. Q. Yang;X. X. Huang

  • Affiliations:

  • Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China;Department of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China

  • Venue:
  • Computational Optimization and Applications
  • Year:
  • 2003

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Abstract

We introduce the concept of partially strictly monotone functions and apply it to construct a class of nonlinear penalty functions for a constrained optimization problem. This class of nonlinear penalty functions includes some (nonlinear) penalty functions currently used in the literature as special cases. Assuming that the perturbation function is lower semi-continuous, we prove that the sequence of optimal values of nonlinear penalty problems converges to that of the original constrained optimization problem. First-order and second-order necessary optimality conditions of nonlinear penalty problems are derived by converting the optimality of penalty problems into that of a smooth constrained vector optimization problem. This approach allows for a concise derivation of optimality conditions of nonlinear penalty problems. Finally, we prove that each limit point of the second-order stationary points of the nonlinear penalty problems is a second-order stationary point of the original constrained optimization problem.