We present new necessary optimality conditions for εe–Pareto optimal solutions of constrained vector optimization problems with empty interior ordering cones. We use the dual-space approach based on advanced tools of variational analysis and generalized differentiation. It allows us not implement any scalarization technique while be able to establish necessary results for nonconvex and nonsolid...

Sparse optimization refers to an optimization problem involving the zero-norm in objective or constraints. In this paper, nonconvex approximation approaches for sparse optimization have been studied with a unifying point of view in DC (Difference of Convex functions) programming framework. Considering a common DC approximation of the zero-norm including all standard sparse inducing penalty func...

We study the estimation of low rank matrices via nonconvex optimization. Compared with convex relaxation, nonconvex optimization exhibits superior empirical performance for large scale instances of low rank matrix estimation. However, the understanding of its theoretical guarantees are limited. In this paper, we define the notion of projected oracle divergence based on which we establish suffic...

Difficult nonconvex optimization problems contain a combinatorial number of local optima, making them extremely challenging for modern solvers. We present a novel nonconvex optimization algorithm that explicitly finds and exploits local structure in the objective function in order to decompose it into subproblems, exponentially reducing the size of the search space. Our algorithm’s use of decom...

We establish theoretical results concerning local optima of regularized M estimators, where both loss and penalty functions are allowed to be nonconvex. Our results show that as long as the loss satisfies restricted strong convexity and the penalty satisfies suitable regularity conditions, any local optimum of the composite objective lies within statistical precision of the true parameter vecto...

To increase the science return of future missions toMars and to enable sample return missions, the accuracy with which a lander can be delivered to the Martian surface must be improved by orders of magnitude. The prior work developed a convex-optimization-based minimum-fuel powered-descent guidance algorithm. In this paper, this convex-optimization-based approach is extended to handle the casew...

In this paper, we generalize the well-known Nesterov’s accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using ...

The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the origi...

Convex optimization has provided both a powerful tool and an intriguing mentality to the analysis and design of communication systems over the last few years. A main challenge today is on nonconvex problems in these application. This paper presents an overview of some of the important nonconvex optimization problems in point-to-point and networked communication systems. Three typical applicatio...