The purpose of this paper is to completely characterize the global approximate optimality (ε-optimality) in reverse convex optimization under the general nonconvex constraint “h(x) ≥ 0″. The main condition presented is obtained in terms of Fenchel’s ε-subdifferentials thanks to El Maghri’s ε-efficiency in difference vector optimization [J. Glob. Optim. 61 (2015) 803–812], after converting the … Read more
Abstract: In a real Hilbert space domain setting, we study the convergence properties of the stochastic Ravine accelerated gradient method for convex differentiable optimization. We consider the general form of this algorithm where the extrapolation coefficients can vary with each iteration, and where the evaluation of the gradient is subject to random errors. This general … Read more
This work considers the effect of averaging, and more generally extrapolation, of the iterates of gradient descent in smooth convex optimization. After running the method, rather than reporting the final iterate, one can report either a convex combination of the iterates (averaging) or a generic combination of the iterates (extrapolation). For several common stepsize sequences, … Read more
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In this paper, we propose a new algorithm for solving monotone mixed variational inequality problems in real Hilbert spaces based on proximal gradient method. Our new algorithm uses a novel explicit stepsize which is proved to be increasing to a positive value. This property plays an important role in improving the speed of the algorithm. … Read more
This paper proposes a stochastic proximal point method to solve a stochastic convex composite optimization problem. High probability results in stochastic optimization typically hinge on restrictive assumptions on the stochastic gradient noise, for example, sub-Gaussian distributions. Assuming only weak conditions such as bounded variance of the stochastic gradient, this paper establishes a low sample complexity … Read more
We focus on constrained, \(L\)-smooth, nonconvex-nonconcave min-max problems either satisfying \(\rho\)-cohypomonotonicity or admitting a solution to the \(\rho\)-weakly Minty Variational Inequality (MVI), where larger values of the parameter \(\rho>0\) correspond to a greater degree of nonconvexity. These problem classes include examples in two player reinforcement learning, interaction dominant min-max problems, and certain synthetic test problems on … Read more
We present a linear cutting-plane relaxation approach that rapidly proves tight lower bounds for the Alternating Current Optimal Power Flow Problem (ACOPF). Our method leverages outer-envelope linear cuts for well-known second-order cone relaxations for ACOPF along with modern cut management techniques. These techniques prove effective on a broad family of ACOPF instances, including the largest … Read more
In this paper, we consider copositive cones over symmetric cones and show that they are never facially exposed when the underlying cone has dimension at least 2. We do so by explicitly exhibiting a non-exposed extreme ray. Our result extends the known fact that the cone of copositive matrices over the nonnegative orthant is not … Read more
We consider (stochastic) subgradient methods for strongly convex but potentially nonsmooth non-Lipschitz optimization. We provide new equivalent dual descriptions (in the style of dual averaging) for the classic subgradient method, the proximal subgradient method, and the switching subgradient method. These equivalences enable $O(1/T)$ convergence guarantees in terms of both their classic primal gap and a … Read more