2020/07/31 by Evelin H. M. Krulikovski, Krulikovski, Evelin H. M., Ademir A. Ribeiro +3
Computer Science · Mathematics · Medicine · #90C30 #90C33 #90C46 #Advanced Optimization Algorithms Research #Cholesterol and Lipid Metabolism #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2008.00019
openalex publication_date 2020/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study a class of optimization problems, called Mathematical\nPrograms with Cardinality Constraints (MPCaC). This kind of problem is\ngenerally difficult to deal with, because it involves a constraint that is not\ncontinuous neither convex, but provides sparse solutions. Thereby we\nreformulate MPCaC in a suitable way, by modeling it as mixed-integer problem\nand then addressing its continuous counterpart, which will be referred to as\nrelaxed problem. We investigate the relaxed problem by analyzing the classical\nconstraints in two cases: linear and nonlinear. In the linear case, we propose\na general approach and present a discussion of the Guignard and Abadie\nconstraint qualifications, proving in this case that every minimizer of the\nrelaxed problem satisfies the Karush-Kuhn-Tucker (KKT) conditions. On the other\nhand, in the nonlinear case, we show that some standard constraint\nqualifications may be violated. Therefore, we cannot assert about KKT points.\nMotivated to find a minimizer for the MPCaC problem, we define new and weaker\nstationarity conditions, by proposing a unified approach that goes from the\nweakest to the strongest stationarity.\n