Nonsmooth Optimization

A Stochastic Proximal-Gradient Method for Regularized Generally Constrained Continuous Optimization

Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization, Stochastic Programming Tags , , , ,

A stochastic proximal-gradient method is proposed and analyzed for solving regularized optimization problems subject to nonlinear equality constraints and nonnegativity bounds. The objective function’s values and gradients are available only through inexact stochastic oracles. The method decomposes each trial step into one component that reduces linearized infeasibility and a second component that seeks optimality while … Read more

Primal-dual forward-reflected splitting with linesearch for distributed min-max problems under local Lipschitz smoothness

Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization

We consider a network of finitely many agents aiming to cooperatively solve a convex-concave min-max problem combining nonsmooth terms with a smooth saddle function whose partial gradients are only locally Lipschitz continuous. Without requiring prior knowledge of the local Lipschitz moduli of the partial gradients, we propose a distributed proximal-reflected-gradient algorithm which incorporates a distributed … Read more

Robust and Distributionally Robust Risk Budgeting with Mean Absolute Deviation

Categories Finance and Economics, Nonsmooth Optimization, Robust Optimization Tags , , ,

We propose robust and distributionally robust formulations of the risk budgeting problem under the mean absolute deviation (MAD) risk measure. The proposed models address uncertainty in asset returns and their distribution while allocating portfolio risk according to prescribed risk budgets. We first develop a robust model in which historical return observations are perturbed within a … Read more

A note on optimality conditions for optimization problems with empty limiting subdifferentials

Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Nonsmooth Optimization

We study first-order optimality for constrained composite optimization problems whose objective is the sum of a locally Lipschitz function and a proper lower semicontinuous function. We focus on the case in which the limiting subdifferential of the latter function is empty at a point of interest, so that the usual KKT conditions are unavailable. We … Read more

Transferability of Error Bounds and the Kurdyka-Lojasiewicz Property under $C^1$ Partial Smoothness

Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

This work studies how a generalized error bound (GEB) property in the ambient Euclidean space transfers to the active manifold under \(\mathcal{C}^1\) partial smoothness. The GEB generalizes the standard error bound, which upper bounds the distance to a subset of critical points using first-order stationarity residuals, by allowing the distance to be composed with a … Read more

A sufficient convergence condition for generalized Benders decomposition with general dual functions

Categories (Mixed) Integer Linear Programming, Convex and Nonsmooth Optimization, Convex Optimization, Nonlinear Optimization, Nonsmooth Optimization, Stochastic Programming Tags , , ,

We revisit the framework of generalized Benders decomposition over a compact but non-finite master domain. We show by counterexample that strong general dual functions alone may fail to guarantee convergence. We then define a condition of uniform local strongness and prove that strong general dual functions satisfying this condition guarantee finite \(\epsilon\)-termination. Finally, we show … Read more

New adaptive proximal gradient algorithms for solving multiobjective composite optimization problems

Categories Nonlinear Optimization, Nonsmooth Optimization, Robust Optimization Tags , , , ,

In this paper, we propose new adaptive proximal gradient algorithms to solve multiobjective optimization problems, where each objective function is the sum of a differentiable function and a proper, closed, convex function. Utilizing the local behavior of the differentiable terms we propose new adaptive ways to select stepsizes used in proximal gradient scheme. In particular, … Read more

A Proximal Approach for Nonsmooth Composite-Constrained Optimization

Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization, Stochastic Programming Tags , , ,

We propose a proximal-type algorithm for nonsmooth and nonconvex optimization problems with composite constraints. The constraint is defined by the composition of a locally upper-\(C^2\) outer function with a locally Lipschitz continuous inner mapping. The method is based on an improvement function that balances objective decrease and constraint satisfaction, and on a surrogate model obtained … Read more

Two Spectral Gaps: Decentralized Optimization over Intersections of Local Convex Sets

Categories Convex Optimization, Nonsmooth Optimization, Parallel Algorithms Tags , , , , ,

We study decentralized minimization of an average of strongly convex, smooth local objectives over an intersection of agent-private closed convex sets, where each agent knows only its own objective and its own set and agents communicate over a gossip network. We show that the complexity is controlled by a single geometric scalar, which we call … Read more

Implicit Primal-Dual Guarantees in Unconstrained First-Order Minimization

Categories Convex Optimization, Nonsmooth Optimization Tags , , , ,

This work considers the design of first-order convex optimization algorithms and convergence proofs. In particular, we consider nonsmooth Lipschitz and smooth problems accessed through a subgradient or gradient oracle, respectively. For the general class of fixed-step first-order methods, prior work on Performance Estimation Problems (PEPs) has shown that structured, tight convergence proofs typically exist. Under … Read more