W propose a new robust optimization method for problems with objective functions that may be computed via numerical simulations and incorporate constraints that need to be feasible under perturbations. The proposed method iteratively moves along descent directions for the robust problem with nonconvex constraints and terminates at a robust local minimum. We generalize the algorithm further to m...

Minibatch decomposition methods for empirical risk minimization are commonly analyzed in a stochastic approximation setting, also known as sampling with replacement. On the other hand, modern implementations of such techniques incremental: they rely on without replacement, which available analysis is much scarcer. We provide convergence guaranties latter variant by analyzing versatile increment...

We propose stochastic optimization algorithms that can find local minima faster than existing algorithms for nonconvex optimization problems, by exploiting the third-order smoothness to escape non-degenerate saddle points more efficiently. More specifically, the proposed algorithm only needs Õ( −10/3) stochastic gradient evaluations to converge to an approximate local minimum x, which satisfies...

In this paper, we are analyzing the performance of energy efficient power allocation for secure orthogonal frequency division multiple access (OFDMA) based cognitive radio networks (CRN‟s). The power allocation schemes are optimized for maximization of the energy efficiency [4] in secure data transmission. The nonconvex optimization problem takes into account to maximize the energy efficiency m...

We provide a unifying geometric framework for the analysis of general classes of duality schemes and penalty methods for nonconvex constrained optimization problems. We present a separation result for nonconvex sets via general concave surfaces. We use this separation result to provide necessary and sufficient conditions for establishing strong duality between geometric primal and dual problems...

We consider a general multiobjective optimization problem with five basic optimality principles: efficiency, weak and proper Pareto optimality, strong efficiency and lexicographic optimality. We generalize the concept of tradeoff directions defining them as some optimal surface of appropriate cones. In convex optimization, the contingent cone can be used for all optimality principles except lex...

Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other high-level search techniques such as branch and bound. However, the conventional duality theory has the fundamental limit that it le...

In this paper we consider nonconvex optimization and learning over a network of distributed nodes. We develop a Proximal Primal-Dual Algorithm (Prox-PDA), which enables the network nodes to distributedly and collectively compute the set of first-order stationary solutions in a global sublinear manner [with a rate of O(1/r), where r is the iteration counter]. To the best of our knowledge, this i...

This paper presents a novel range division and contraction approach for globally solving nonconvex quadratic program with quadratic constraints. By constructing new underestimating linear relaxation functions, we can transform the initial nonconvex quadratic program problem into a linear program relaxation problem. By employing a branch and bound scheme with a range contraction approach, we des...