We introduce a hybrid projection-localization method for solving large convex cone programs. The method interleaves a series of projections onto localization sets with the classical alternating projections method for convex feasibility problems; the problem is made amenable to the method by reducing it to a convex feasibility problem composed of a subspace and a cone via its homogeneous self-du...

In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions. We prove, under natural assumptions, that the sequence generated by the method is well defined and converges globally to a Pareto-Clarke critical point. Our method may be seen as an extension, for the non convex case,...

The Alternating Direction Method of Multipliers (ADMM) is a method that solves convex optimization problems of the form min(f(x) + g(z)) subject to Ax + Bz = c, where A and B are suitable matrices and c is a vector, for optimal points (xopt, zopt). It is commonly used for distributed convex minimization on large scale data-sets. However, it can be technically difficult to implement and there is...

This paper is concerned with the alternating minimization (AM) method for solving convex minimization problems where the decision variables vector is split into two blocks. The objective function is a sum of a differentiable convex function and a separable (possibly) nonsmooth extended real-valued convex function, and consequently constraints can be incorporated. We analyze the convergence rate...

We study discrete optimization problems with a submodular mean-risk minimization objective. For 0-1 problems a linear characterization of the convex lower envelope is given. For mixed 0-1 problems we derive an exponential class of conic quadratic inequalities. We report computational experiments on risk-averse capital budgeting problems with uncertain returns.

It is well known that the gradient-projection algorithm plays an important role in solving constrained convex minimization problems. In this paper, based on Xu’s method [Xu, H. K.: Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl. 150, 360-378(2011)], we use the idea of regularization to establish implicit and explicit iterative methods for finding the approximate ...

Convex loss minimization with l1 regularization has been proposed as a promising method for feature selection in classification (e.g., l1-regularized logistic regression) and regression (e.g., l1-regularized least squares). In this paper we describe an efficient interior-point method for solving large-scale l1-regularized convex loss minimization problems that uses a preconditioned conjugate gr...

We provide a comprehensive study of the convergence forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe fine geometry objective functions relevant in inverse problems signal processing, that have nice behaviour on manifolds, sets open with respect weak topo...