On solving convex optimization problems with linear ascending constraints

Abstract

In this paper, we propose two algorithms for solving convex optimization problems with linear ascending constraints. When the objective function is separable, we propose a dual method which terminates in a finite number of iterations. In particular, the worst case complexity of our dual method improves over the best-known result for this problem in Padakandla and Sundaresan (SIAM J Optim 20 (3): [1185][1186][1187][1188][1189][1190][1191][1192][1193][1194][1195][1196][1197][1198][1199][1200][1201][1202][1203][1204] 2009). We then propose a gradient projection method to solve a more general class of problems in which the objective function is not necessarily separable. Numerical experiments show that both our algorithms work well in test problems.

Convex separable problems with linear and box constraints

“…The main related literature to this paper is represented by [5], [10] and [11]. In [5], the authors propose a general framework for solving optimization problems in which the solutions have a waterfilling structure.…”

“…In [5], the authors propose a general framework for solving optimization problems in which the solutions have a waterfilling structure. In [10] and [11], the authors focus on solving problems of the form:…”

“…In particular, in [10] the authors propose a dual method to numerically evaluate the solution of the above problem in no more than N − 1 iterations under an ordering condition on the slopes of the functions at the origin. An alternative solution improving the worst case complexity of [10] is illustrated in [11]. Differently from [10] and [11], we consider more general problems in which the inequality constraints are not necessarily in ascending order and the box constraint values l n and u n may possibly be equal to −∞ and +∞, respectively.…”

“…An alternative solution improving the worst case complexity of [10] is illustrated in [11]. Differently from [10] and [11], we consider more general problems in which the inequality constraints are not necessarily in ascending order and the box constraint values l n and u n may possibly be equal to −∞ and +∞, respectively. All this makes (1) more general than problems of the form given in (2).…”

“…All this makes (1) more general than problems of the form given in (2). Also, some of the restrictions on functions f n that were present in [10] and [11] have been removed 1 .…”