Sequential optimality conditions for cardinality-constrained optimization problems with applications

Abstract

Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka–Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.

A New Sequential Optimality Condition of Cardinality-Constrained Optimization Problem and Application

“…in other words, CC-PAM-regularity condition is a CC-CQ. Literature [18] calls the constraint qualification that satisfies the property (11) as the strict constraint qualification (SCQ). And the conclusion (iii) means that the CC-PAMregularity condition is the weakest SCQ relative to CC-PAM-stationarity.…”

“…• We define a new problem-tailored constraint qualification, called CC-PAMregularity, which is weaker than CC-AM-regularity proposed in [11]. We prove that any CC-M-stationary point is CC-PAM-stationary, and conversely, CC-PAM-stationary point is CC-M-stationary if CC-PAM-regularity condition is satisfied.…”

“…• We apply CC-PAM-stationarity to safeguarded augmented Lagrangian method and further improve its convergence. Different from the regularization methods, the literature [9] and [11] try to directly apply the safeguarded augmented Lagrangian method of the general NLP to the relaxation problem and [9] uses the corresponding solver ALGENCAN [18][19][20] to solve the portf olio problem verify the advantages of the augmented Lagrangian algorithm over the regularization methods. In addition, the above two regularization methods both require accurate KKT points for their subproblems, while the safeguarded augmented Lagrangian method only requires subproblems to be solved inaccurately.…”

“…In addition, Kanzow et al [9] adapted the quasi-normality CQ in [10] and obtained a form corresponding to CCOP. And [11] proposed a cone-continuity constraint qualification. At the same time, [7] defines the first-order stationarity concept of the relaxation problem, called CC-Strong-stationary (CC-S-stationary) and CC-Mordukhovich-stationary (CC-M-stationary), where CC-S-stationary is equivalent to the Karush-Kuhn-T ucker (KKT) condition of the relaxation problem, and the CC-M-stationary is equivalent to the KKT condition of T N LP (x * ); [12] provides a Weak-type stationarity.…”

“…[12] establishes a sequential optimality condition, called CC-Approximate Weak stationarity (CC-AW-stationarity), but this condition is based on the (x, y) space. Therefore, Kanzow et al [11] proposed CC-Approximate Mordukhovich stationarity (CC-AM-stationarity), which is only related to x, and a proof is given that it is equivalent to CC-AW-stationary. However, for some problems, the number of CC-AM-stationary points is numerous, and these points are often far from the optimal solutions (e.g.…”