Convergence analysis for the modified spectral projected subgradient method

Abstract

In a recent paper, the spectral projected subgradient (SPS) method was introduced by Loreto et al. for the minimization of a non-differentiable convex piecewise function, and extensive numerical experimentation showed that this method was very efficient. However, no theoretical convergence was shown. In this paper, a modified version of the spectral projected subgradient (MSPS) is presented. The MSPS is the result of applying to SPS the direction approach used by spectral projected gradient version one (SPG1) proposed by Raydan et al. MSPS presents stronger convergence properties than SPS. We give a comprehensive theoretical analysis of the MSPS and its convergence is shown under some mild assumptions. The proof uses the traditional scheme of descent distance to the optimal value set, and a non-monotone globalization condition is used to get that distance instead of the subgradient definition. To illustrate the behavior of MSPS we present and discuss numerical results for set covering problems.

A subgradient method with non-monotone line search

“…which shows that our method is different from the ones that appeared in [15][16][17]. Moreover, our line search allows different choices for the sequence (γ k ) k∈N that controls the non-monotonicity.…”

“…It is worth to noting that the idea of using general nonmonotone line searches in differentiable optimization, generalizing the non-monotone searches proposed in [10,24], have appeared in [9,22]. A modified version of the subgradient method with the non-monotone line search proposed in [10] was considered in [15][16][17].…”

A subgradient method with non-monotone line search

“…It is worth to noting that the idea of using general non-monotone line searches in differentiable optimization, generalizing the non-monotone searches proposed in [12,27], have appeared in [11,24]. A modified version of the subgradient method with the non-monotone line search proposed in [12] was considered in [17][18][19], see also [14].…”

“…They show that the proposed non-monotone method compares favorably with the classical subgradient method endowed with usual prefixed s tep s izes. I t i s worth m entioning t hat o ther modified versions of the subgradient method with non-monotonic linear search proposed in [12] were considered in [17][18][19], see also [14].…”

A Water Fraction Measurement Method Using Heuristic-Algorithm-Based Electrical Capacitance Tomography Images Post-Processing Technology

“…To accelerate the convergence of ( 7), reference [18] indicated that Barzilai and Borwein Gradient method [19], [20] can be used for the updating of parameter τ in the k-th iteration…”