2018/08/23 by Tatiana Tatarenko, Tatarenko, Tatiana, Angelia Nedich +1
Computer Science · Engineering · Medicine · #Aortic aneurysm repair treatments #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1808.07749
openalex publication_date 2018/08/23 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
In this work, we consider a constrained convex problem with linear\ninequalities and provide an inexact penalty re-formulation of the problem. The\nnovelty is in the choice of the penalty functions, which are smooth and can\ninduce a non-zero penalty over some points in feasible region of the original\nconstrained problem. The resulting unconstrained penalized problem is\nparametrized by two penalty parameters which control the slope and the\ncurvature of the penalty function. With a suitable selection of these penalty\nparameters, we show that the solutions of the resulting penalized unconstrained\nproblem are \feasible for the original constrained problem, under some\nassumptions. Also, we establish that, with suitable choices of penalty\nparameters, the solutions of the penalized unconstrained problem can achieve a\nsuboptimal value which is arbitrarily close to the optimal value of the\noriginal constrained problem. For the problems with a large number of linear\ninequality constraints, a particular advantage of such a smooth penalty-based\nreformulation is that it renders a penalized problem suitable for the\nimplementation of fast incremental gradient methods, which require only one\nsample from the inequality constraints at each iteration. We consider applying\nSAGA proposed in citesaga to solve the resulting penalized unconstrained\nproblem. Moreover, we propose an alternative approach to set up the penalized\nproblem. This approach is based on the time-varying penalty parameters and,\nthus, does not require knowledge about some problem-specific properties, that\nmight be difficult to estimate. We prove that the single-loop full\ngradient-based algorithm applied to the corresponding time-varying penalized\nproblem converges to the solution of the original constrained problem in the\ncase of the strongly convex objective function.\n