A global regularity theorem for stress fields which correspond to minimisers of convex and some special nonconvex variational problems is derived for Lipschitz-domains. In the first part it is assumed that the energy densities defining the variational problem are convex but not necessarily strictly convex and satisfy a convexity inequality. The regularity result for this case is derived with a ...

A branch and bound global optimization method, BB, for general continuous optimization problems involving nonconvexities in the objective function and/or constraints is presented. The nonconvexities are categorized as being either of special structure or generic. A convex relaxation of the original nonconvex problem is obtained by (i) replacing all nonconvex terms of special structure (i.e. bil...

We study the low rank matrix factorization problem via nonconvex optimization. Compared with the convex relaxation approach, nonconvex optimization exhibits superior empirical performance for large scale low rank matrix estimation. However, the understanding of its theoretical guarantees is limited. To bridge this gap, we exploit the notion of inexact first order oracle, which naturally appears...

Motivated by the fact that important real-life problems, such as the protein docking problem, can be accurately modeled by minimizing a nonconvex piecewise-quadratic function, a nonconvex underestimator is constructed as the minimum of a finite number of strictly convex quadratic functions. The nonconvex underestimator is generated by minimizing a linear function on a reverse convex region and ...

Many contemporary signal processing, machine learning and wireless communication applications can be formulated as nonconvex nonsmooth optimization problems. Often there is a lack of efficient algorithms for these problems, especially when the optimization variables are nonlinearly coupled in some nonconvex constraints. In this work, we propose an algorithm named penalty dual decomposition (PDD...

This paper presents a methodology in the Linear Matrix Inequality (LMI) framework to jointly optimize the linear control law and the linear parameters in the plant. The paper solves an integrated plant and control design problem which bounds the covariance of selected outputs. The method simultaneously designs the plant parameters and the controller. The proposed method also allows one to guara...

In a number of application areas, it is desirable to obtain sparse solutions. Minimizing the number of nonzeroes of the solution (its `0-norm) is a difficult nonconvex optimization problem, and is often approximated by the convex problem of minimizing the `1-norm. In contrast, we consider exact formulations as mathematical programs with complementarity constraints and their reformulations as sm...

We develop a spatial branch-and-cut approach for nonconvex Quadratically Constrained Quadratic Programs with bounded complex variables (CQCQP). Linear valid inequalities are added at each node of the search tree to strengthen semidefinite programming relaxations of CQCQP. These valid inequalities are derived from the convex hull description of a nonconvex set of 2 × 2 positive semidefinite Herm...

The power systems state estimation problem computes the set of complex voltage phasors given quadratic measurements using nonlinear least squares (NLS). This is a nonconvex optimization problem, so even in the absence of measurement errors, local search algorithms like Newton / Gauss–Newton can become “stuck” at local minima, which correspond to nonsensical estimations. In this paper, we observ...