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.
Highlights
We consider cardinality-constrained (CC) optimization problems of the form minx f (x) s.t. g(x) ≤ 0, h(x) = 0, ‖x‖0 ≤ s, (1.1)where f ∈ C1(Rn, R), g ∈ C1(Rn, Rm), h ∈ C1(Rn, Rp), and ‖x‖0 denotes the number of nonzero components of a vector x
We first derive a sequential optimality condition called CC-AM-stationarity for (1.1), which is the CC-analogue of the approximate Karush-Kuhn-Tucker (AKKT) condition for standard nonlinear optimization problems (NLP) introduced in [3, 10, 25], see [6, 26] for similar concepts in the context of MPCCs. We show that this first-order necessary optimality condition is satisfied at every local minimizer of (1.1) without requiring a constraint qualification
In order to establish the relationship between CC-AM-stationarity and the CCM-stationarity condition introduced in [12, 31], we propose a constraint qualification called CC-AM-regularity based on a cone-continuity property
Summary
The regularization method from [17] is adapted to solve the reformulated problem and it is shown that any limit point of this method satisfies the CC-M-stationarity condition provided that a constraint qualification called CC-CPLD holds at this limit point. This convergence result is only proven for the exact case, i.e., under the assumption that an exact KKT point of the regularized subproblem can be computed in each iteration. We write Br(x) and Br(x) for an open and closed ball with radius r > 0 around x
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