In this paper we consider the problem of minimizing a (possibly nonconvex) quadratic function with a quadratic constraint. We point out some new properties of the problem. In particular, in the rst part of the paper, we show that (i) given a KKT point that is not a global minimizer, it is easy to nd a \better" feasible point; (ii) strict complementarity holds at the local-nonglobal minimizer. I...

Concave-Convex Procedure (CCCP) has been widely used to solve nonconvex d.c.(difference of convex function) programs occur in learning problems, such as sparse support vector machine (SVM), transductive SVM, sparse principal componenent analysis (PCA), etc. Although the global convergence behavior of CCCP has been well studied, the convergence rate of CCCP is still an open problem. Most of d.c....

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...

We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gradient method with constant minibatch converges to a stationary point. To tackle this issue, we develop fast st...

We consider the problem of computing a critical point of a nonconvex locally Lipschitz function over a convex compact constraint set given an inexact oracle that provides an approximate function value and an approximate subgradient. We assume that the errors in function and subgradient evaluations are merely bounded, and in particular need not vanish in the limit. After some discussion on how t...

The success of deep learning has led to a rising interest in the generalization property of the stochastic gradient descent (SGD) method, and stability is one popular approach to study it. Existing works based on stability have studied nonconvex loss functions, but only considered the generalization error of the SGD in expectation. In this paper, we establish various generalization error bounds...

Convex representatives are proposed for the value function of an infinite-dimensional constrained nonconvex variational problem. All involved variables in this problem take their values (possibly infinite dimension, not necessarily separable or complete) normed spaces, while associated measure can be any $\sigma$-finite, nonnegative, and nonatomic complete measure. This particular shows that cl...

The problem of cluster analysis is formulated as a problem of nonsmooth, nonconvex optimization. An algorithm for solving the latter optimization problem is developed which allows one to significantly reduce the computational efforts. This algorithm is based on the so-called discrete gradient method. Results of numerical experiments are presented which demonstrate the effectiveness of the propo...

In this paper, we study second-order necessary optimality conditions for a discrete optimal control problem with nonconvex cost functions and state-control constraints. By establishing an abstract result on second-order necessary optimality conditions for a mathematical programming problem, we derive second-order necessary optimality conditions for a discrete optimal control problem.

This paper presents a pure complementary energy variational method for solving a general anti-plane shear problem in finite elasticity. Based on the canonical dualitytriality theory developed by the author, the nonlinear/nonconex partial differential equations for the large deformation problem is converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a ...