In this paper we study smooth convex programming problems where the decision variables vector is split into several blocks of variables. We analyze the block coordinate gradient projection method in which each iteration consists of performing a gradient projection step with respect to a certain block taken in a cyclic order. Global sublinear rate of convergence of this method is established and it is shown that it can be accelerated when the problem is unconstrained. In the unconstrained setting we also prove a sublinear rate of convergence result for the so-called alternating minimization method when the number of blocks is two. When the objective function is also assumed to be strongly convex, linear rate of convergence is established. Introduction.One of the first variable decomposition methods for solving general minimization problems is the so-called alternating minimization method [5,14], which is based on successive global minimization with respect to each component vector in a cyclic order. This fundamental method appears in the literature under various names such as the block-nonlinear Gauss-Seidel method or the block coordinate descent method (see, e.g., [4]). The convergence of the method was extensively studied in the literature under various assumptions. For example, Auslender studied in [1] the convergence of the method under a strong convexity assumption, but without assuming differentiability. In [4] Bertsekas showed that if the minimum with respect to each block of variables is unique, then any accumulation point of the sequence generated by the method is also a stationary point. Grippo and Sciandrone showed in [7] convergence results of the sequence generated by the method under different sets of assumptions such as strict quasi convexity with respect to each block. Luo and Tseng proved in [9] that under the assumptions of strong convexity with respect to each block, existence of a local error bound of the objective function, and proper separation of isocost surfaces, linear rate of convergence can be established.Another closely related method, which will be the main focus of this paper, is the block coordinate gradient projection (BCGP) method in which at each subiteration, the exact minimization with respect to a certain block of variables is replaced with an employment of a single step of the gradient projection method (a step toward the gradient followed by an orthogonal projection). This method has a clear advantage over alternating minimization when exact minimization with respect to each of the

Nonconvex optimization, Successive convex approximations, KKT points, Truss topology design, Displacement and stress constraints,

The projected subgradient method for constrained minimization repeatedly interlaces subgradient steps for the objective function with projections onto the feasible region, which is the intersection of closed and convex constraints sets, to regain feasibility. The latter poses a computational difficulty and, therefore, the projected subgradient method is applicable only when the feasible region is "simple to project onto". In contrast to this, in the superiorization methodology a feasibility-seeking algorithm leads the overall process and objective function steps are interlaced into it. This makes a difference because the feasibility-seeking algorithm employs projections onto the individual constraints sets and not onto the entire feasible region.We present the two approaches side-by-side and demonstrate their performance on a problem of computerized tomography image reconstruction, posed as a constrained minimization problem aiming at finding a constraint-compatible solution that has a reduced value of the total variation of the reconstructed image.

Abstract-We propose a distributed positioning algorithm to estimate the unknown positions of a number of target nodes, given distance measurements between target nodes and between target nodes and a number of reference nodes at known positions. Based on a geometric interpretation, we formulate the positioning problem as an implicit convex feasibility problem in which some of the sets depend on the unknown target positions, and apply a parallel projection onto convex sets approach to estimate the unknown target node positions. The proposed technique is suitable for parallel implementation in which every target node in parallel can update its position and share the estimate of its location with other targets. We mathematically prove convergence of the proposed algorithm. Simulation results reveal enhanced performance for the proposed approach compared to available techniques based on projections, especially for sparse networks.Index Terms-Positioning, Cooperative wireless sensor network, Parallel projections onto convex sets, Convex feasibility problem, Implicit convex feasibility.

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