In what follows, we provide the demand analysis associated with budget constrained linear utility maximization for each of several categories of goods, with the marginal rate of consumption expenditure-as a share of wealth- being a positive constant less than one. The marginal rate of consumption expenditure is endogenously determined, by a budget constrained “Cobb-Douglas with … Read more
We propose a new approach based DC programming for fnding a solution of the partial facility interdiction problem that belongs to the class of bilevel programming. This model was frst considered in the work of Aksen et al. [1] with a heuristic algorithm named multi-start simplex search (MSS). However, because of the big number of … Read more
This paper focuses on solving mathematical programs with complementarity constraints (MPCCs) without assuming MPCC-LICQ or lower level strict complementarity at a solution. We show that a local minimizer of an MPCC is “piecewise M-stationary” un- der MPCC-GCQ; furthermore, every weakly stationary point of an MPCC is B-stationary if MPCC-ACQ holds. For the Bounding Algorithm proposed … Read more
We present Zeroth-order Riemannian Averaging Stochastic Approximation (\texttt{Zo-RASA}) algorithms for stochastic optimization on Riemannian manifolds. We show that \texttt{Zo-RASA} achieves optimal sample complexities for generating $\epsilon$-approximation first-order stationary solutions using only one-sample or constant-order batches in each iteration. Our approach employs Riemannian moving-average stochastic gradient estimators, and a novel Riemannian-Lyapunov analysis technique for convergence analysis. … Read more
We prove the first convergence guarantees for a subgradient method minimizing a generic Lipschitz function over generic Lipschitz inequality constraints. No smoothness or convexity (or weak convexity) assumptions are made. Instead, we utilize a sequence of recent advances in Lipschitz unconstrained minimization, which showed convergence rates of $O(1/\delta\epsilon^3)$ towards reaching a “Goldstein” stationary point, that … Read more
Stochastic sequential quadratic optimization (SQP) methods for solving continuous optimization problems with nonlinear equality constraints have attracted attention recently, such as for solving large-scale data-fitting problems subject to nonconvex constraints. However, for a recently proposed subclass of such methods that is built on the popular stochastic-gradient methodology from the unconstrained setting, convergence guarantees have been … Read more
Restricted normal cones are of interest, for instance, in the theory of local error bounds, where they have recently been used to characterize the exis- tence of a constrained Lipschitzian error bound. In this paper, we establish rela- tions between two concepts for restricted normals. The first of these concepts was introduced in the late … Read more
In the past years, augmented Lagrangian methods have been successfully applied to several classes of non-convex optimization problems, inspiring new developments in both theory and practice. In this paper we bring most of these recent developments from nonlinear programming to the context of optimization on Riemannian manifolds, including equality and inequality constraints. Many research have … Read more
Bilevel optimization problems are known to be challenging to solve in practice. In particular, the feasible set of a bilevel problem is, in general, non-convex, even for linear bilevel problems. In this work, we aim to develop a better understanding of the feasible set of linear bilevel problems. Specifically, we develop means by which to … Read more
We explore Maximum a Posteriori inference of Bayesian Hierarchical Models (BHMs) with intractable normalizers, which are increasingly prevalent in contemporary applications and pose computational challenges when combined with nonconvexity and nondifferentiability. To address these, we propose the Adaptive Importance Sampling-based Surrogation method, which efficiently handles nonconvexity and nondifferentiability while improving the sampling approximation of the … Read more