We propose two numerical algorithms for minimizing the sum of a smooth function and the composition of a nonsmooth function with a linear operator in the fully nonconvex setting. The iterative schemes are formulated in the spirit of the proximal and, respectively, proximal linearized alternating direction method of multipliers. The proximal terms are introduced through … Read more
This paper concerns with the group zero-norm regularized least squares estimator which, in terms of the variational characterization of the zero-norm, can be obtained from a mathematical program with equilibrium constraints (MPEC). By developing the global exact penalty for the MPEC, this estimator is shown to arise from an exact penalization problem that not only … Read more
In the paper [11] we derived first order (KKT) and second order (SSC) optimality conditions for functions defined by evaluation programs involving smooth elementals and absolute values. For this class of problems we showed in [12] that the natural algorithm of successive piecewise linear optimization with a proximal term (SPLOP) achieves a linear or even … Read more
Proximal gradient methods have been found to be highly effective for solving minimization problems with non-negative constraints or L1-regularization. Under suitable nondegeneracy conditions, it is known that these algorithms identify the optimal sparsity pattern for these types of problems in a finite number of iterations. However, it is not known how many iterations this may … Read more
An abstract convergence theorem for a class of generalized descent methods that explicitly models relative errors is proved. The convergence theorem generalizes and unifies several recent abstract convergence theorems. It is applicable to possibly non-smooth and non-convex lower semi-continuous functions that satisfy the Kurdyka–Lojasiewicz (KL) inequality, which comprises a huge class of problems. Most of … Read more
Abstract. In the paper [9] we derived first order (KKT) and second order (SSC) optimality conditions for functions defined by evaluation programs involving smooth elementals and absolute values. In that analysis, a key assumption on the local piecewise linearization was the Linear Independence Kink Qualification (LIKQ), a generalization of the Linear Independence Constraint Qualification (LICQ) … Read more
We consider a popular nonsmooth formulation of the real phase retrieval problem. We show that under standard statistical assumptions, a simple subgradient method converges linearly when initialized within a constant relative distance of an optimal solution. Seeking to understand the distribution of the stationary points of the problem, we complete the paper by proving that … Read more
Functions defined by evaluation programs involving smooth elementals and absolute values as well as the max- and min-operator are piecewise smooth. Using piecewise linearization we derived in [7] for this class of nonsmooth functions first and second order conditions for local optimality (MIN). They are necessary and sufficient, eespectively. These generalizations of the classical KKT … Read more
We study the worst-case convergence rates of the proximal gradient method for minimizing the sum of a smooth strongly convex function and a non-smooth convex function whose proximal operator is available. We establish the exact worst-case convergence rates of the proximal gradient method in this setting for any step size and for different standard performance … Read more
An algorithm framework is proposed for minimizing nonsmooth functions. The framework is variable-metric in that, in each iteration, a step is computed using a symmetric positive definite matrix whose value is updated as in a quasi-Newton scheme. However, unlike previously proposed variable-metric algorithms for minimizing nonsmooth functions, the framework exploits self-correcting properties made possible through … Read more