Global convergence and finite termination of a class of smooth penalty function algorithms

Abstract

In this paper, based on a new class of asymptotic l 1 exact penalty functions, we propose a smooth penalty function for solving nonlinear programming problems. One of the main features of our algorithm is that at each iteration, we do not need to solve the global minimum of penalty functions. Furthermore, global convergence property is established without requiring any constraint qualifications. By addressing perturbation functions, we obtain that the lower semi-continuity of the perturbation function at zero is a necessary and sufficient condition to ensure the convergence of the objective function values generated by the algorithm to the optimal value of the primal problem. Since the perturbation function is only dependent on the data of the primal problem, these results enable us to check the convergence property of the algorithm in advance. Finally, we discuss the finite termination of the algorithm when the solution set is non-degenerate or weakly sharp.

The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems

“…Remark 8. In smooth problems, a special case of EP(ϕ, S), Corollary 10 is just (Theorem 5.3 [33]), and the latter is an extension of (Corollary 3.5 [20]).…”

“…By Theorem 2 and a series of its corollaries, one can see that, under normal conditions, the weak sharpness or strong non-degeneracy of the solution set is a special case of the augmented weak sharpness with respect to the feasible solution sequence. On the other hand, for some algorithms in mathematical programming and variational inequalities, for example, the proximal point algorithm, the gradient projection algorithm and the SQP algorithm, and so on (see [18,20,[32][33][34]38,39]), the projected gradient of the point sequence generated by them all converge to zero, i.e., (50) holds. Therefore, the notion of augmented weak sharpness of the solution set presented by us provides weaker sufficient conditions than the weak sharpness or strong non-degeneracy for the finite termination of these algorithms.…”

“…To extend these characteristics to variational inequalities and smooth non-convex programming problems, several studies ( [22,[32][33][34][35]) have employed (9) to define the weak sharpness of the solution set in these contexts. We now propose to use (8) to define the weak sharpness of the solution set S for EP(ϕ, S).…”

Finite Convergence for Feasible Solution Sequence of Variational Inequality Problems

“…Finite convergence of the feasible solution sequence produced by any algorithm for VIP has been widely concerned and researched for a long time. Early in previous studies, Rockafellar [1], Polyak [2], and Ferris [3] successively put forward the weak sharp minima and strong non-degeneracy of the solution sets of mathematical programming problems, and prove that any one of them is the sufficient condition for finite convergence of the proximal point algorithm [1,[4][5][6] and some important iterative algorithms [7][8][9][10][11][12]. It is worth noting that the finite convergence of the feasible solution sequence produced by the algorithms above, depending on not only the weak sharp minima or strong non-degeneracy of the solution set, but also the specific structural features of the algorithms.…”

Smoothing Approximation to the Square-Order Exact Penalty Functions for Constrained Optimization

“…However, it is not a smooth function and causes some numerical instability problems in its implementation when the value of the penalty parameter becomes larger. Some methods for smoothing the exact penalty function are developed (see, e.g., [7][8][9][10][11][12][13][14]). In [15,16], the square-order penalty function…”