In this two-part paper, we propose a general algorithmic framework for the minimization of a nonconvex smooth function subject to nonconvex smooth constraints. The algorithm solves a sequence of (separable) strongly convex problems and mantains feasibility at each iteration. Convergence to a stationary solution of the original nonconvex optimization is established. Our framework is very general...

We suggest a branch and bound algorithm for solving continuous optimization problems where a (generally nonconvex) objective function is to be minimized under nonconvex inequality constraints which satisfy some specific solvability assumptions. The assumptions hold for some special cases of nonconvex quadratic optimization problems. We show how the algorithm can be applied to the problem of min...

We describe a primal-dual application of the proximal point algorithm to nonconvex minimization problems. Motivated by the work of Spingarn and more recently by the work of Kaplan and Tichatschke about the proximal point methodology in nonconvex optimization. This paper discusses some local results in two directions. The first one concerns the application of the proximal method of multipliers t...

We take a Hamiltonian-based perspective to generalize Nesterov's accelerated gradient descent and Polyak's heavy ball method broad class of momentum methods in the setting (possibly) constrained minimization Euclidean non-Euclidean normed vector spaces. Our leads generic unifying nonasymptotic analysis convergence these both function value (in convex optimization) norm unconstrained, possibly n...

The paper identifies classes of nonconvex optimization problems whose convex relaxations have optimal solutions which at the same time are global optimal solutions of the original nonconvex problems. Such a hidden convexity property was so far limited to quadratically constrained quadratic problems with one or two constraints. We extend it here to problems with some partial separable structure....

In this paper, we study the stochastic gradient descent (SGD) method for the nonconvex nonsmooth optimization, and propose an accelerated SGD method by combining the variance reduction technique with Nesterov’s extrapolation technique. Moreover, based on the local error bound condition, we establish the linear convergence of our method to obtain a stationary point of the nonconvex optimization....

This paper addresses the problem of integrating in a single model operational optimization and cyclic scheduling of continuous plants. Considered are multiproduct, multistage plants with finite intermediate storage capacity (FIS). A combined optimization approach introduces synergic effects for more effective scheduling and plant operation (Alle and Pinto, 2001a,b). The representation proposed ...

Accurate nonlinear dynamic models of process operations such as start-ups, shut-downs, and complex changeovers include state dependent events that trigger discrete changes to the describing equations, and are best analyzed within a hybrid systems framework. The automated design of an optimal process operation can thus be formulated as a dynamic optimization problem with a hybrid system embedded...

In this paper, the estimation problem for sparse reduced rank regression (SRRR) model is considered. The SRRR model is widely used for dimension reduction and variable selection with applications in signal processing, econometrics, etc. The problem is formulated to minimize the least squares loss with a sparsity-inducing penalty considering an orthogonality constraint. Convex sparsity-inducing ...

In this paper we analyze several new methods for solving nonconvex optimization problems with the objective function formed as a sum of two terms: one is nonconvex and smooth, and another is convex but simple and its structure is known. Further, we consider both cases: unconstrained and linearly constrained nonconvex problems. For optimization problems of the above structure, we propose random ...