On the Constant Positive Linear Dependence Condition and Its Application to SQP Methods

In this paper, we introduce a constant positive linear dependence condition (CPLD), which is weaker than the Mangasarian--Fromovitz constraint qualification (MFCQ) and the constant rank constraint qualification (CRCQ). We show that a limit point of a sequence of approximating Karush--Kuhn--Tucker (KKT) points is a KKT point if the CPLD holds there. We show that a KKT point satisfying the CPLD and the strong second-order sufficiency conditions (SSOSC) is an isolated KKT point. We then establish convergence of a general sequential quadratical programming (SQP) method under the CPLD and the SSOSC. Finally, we apply these results to analyze the feasible SQP method proposed by Panier and Tits in 1993 for inequality constrained optimization problems. We establish its global convergence under the SSOSC and a condition slightly weaker than the Mangasarian--Fromovitz constraint qualification, and we prove superlinear convergence of a modified version of this algorithm under the SSOSC and a condition slightly weaker than the linear independence constraint qualification.

[1]  Jong-Shi Pang,et al.  Piecewise Smoothness, Local Invertibility, and Parametric Analysis of Normal Maps , 1996, Math. Oper. Res..

[2]  Stephen J. Wright,et al.  Superlinear convergence of an interior-point method for monotone variational inequalities , 1996 .

[3]  Liqun Qi,et al.  Superlinearly convergent approximate Newton methods for LC1 optimization problems , 1994, Math. Program..

[4]  Liqun Qi,et al.  An SQP algorithm for extended linear-quadratic problems in stochastic programming , 1995, Ann. Oper. Res..

[5]  J. Pang,et al.  Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory , 1994 .

[6]  Olvi L. Mangasarian,et al.  Superlinearly convergent quasi-newton algorithms for nonlinearly constrained optimization problems , 1976, Math. Program..

[7]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[8]  S. M. Robinson.  Generalized equations and their solutions, part II: Applications to nonlinear programming , 1982 .

[9]  Stephen J. Wright.  Superlinear Convergence of a Stabilized SQP Method to a Degenerate Solution , 1998, Comput. Optim. Appl..

[10]  E. Panier,et al.  A superlinearly convergent feasible method for the solution of inequality constrained optimization problems , 1987 .

[11]  Olvi L. Mangasarian,et al.  Nonlinear Programming , 1969 .

[12]  J. F. Bonnans.  Local study of Newton type algorithms for constrained problems , 1988 .

[13]  Shih-Ping Han,et al.  Superlinearly convergent variable metric algorithms for general nonlinear programming problems , 1976, Math. Program..

[14]  André L. Tits,et al.  On combining feasibility, descent and superlinear convergence in inequality constrained optimization , 1993, Math. Program..

[15]  Xiaojun Chen,et al.  Convergence of the BFGS Method for LC 1 Convex Constrained Optimization , 1996 .

[16]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

[17]  Andreas Fischer,et al.  Modified Wilson's Method for Nonlinear Programs with Nonunique Multipliers , 1999, Math. Oper. Res..

[18]  J. Stoer,et al.  Convexity and Optimization in Finite Dimensions I , 1970 .

[19]  Defeng Sun,et al.  Superlinear Convergence of Approximate Newton Methods for LC1 Optimization Problems without Strict Complementarity , 1995 .

[20]  Houyuan Jiang,et al.  Semismooth Karush-Kuhn-Tucker Equations and Convergence Analysis of Newton and Quasi-Newton Methods for Solving these Equations , 1997, Math. Oper. Res..

[21]  Houyuan Jiang,et al.  Globally and superlinearly convergent trust-region algorithm for convex SC1-minimization problems and its application to stochastic programs , 1996 .

[22]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[23]  J. Pang,et al.  A globally convergent Newton method for convex SC1 minimization problems , 1995 .

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

[25]  Stephen M. Robinson,et al.  Lower Semicontinuity of Multivalued Linearization Mappings , 1973 .

[26]  Francisco Facchinei,et al.  Minimization of SC1 functions and the Maratos effect , 1995, Oper. Res. Lett..

[27]  M. J. D. Powell,et al.  THE CONVERGENCE OF VARIABLE METRIC METHODS FOR NONLINEARLY CONSTRAINED OPTIMIZATION CALCULATIONS , 1978 .

[28]  L. Qi,et al.  Newton's method for quadratic stochastic programs with recourse , 1995 .

[29]  S. M. Robinson.  First Order Conditions for General Nonlinear Optimization , 1976 .

[30]  Elijah Polak,et al.  Globally and Superlinearly Convergent Algorithm for Minimizing a Normal Merit Function , 1998 .

[31]  M. Kojima.  Strongly Stable Stationary Solutions in Nonlinear Programs. , 1980 .

[32]  Houyuan Jiang,et al.  Semismoothness and Superlinear Convergence in Nonsmooth Optimization and Nonsmooth Equations , 1996 .