lipschitz continuity – Optimization Online

A symmetric extrapolated proximal alternating predictor-corrector method for saddle-point problems

Published: 2025/05/28

Convex Optimization lipschitz continuity, primal-dual algorithms, Proximal alternating predictor-corrector, saddle point problem, symmetry

The proximal alternating predictor-corrector (PAPC) method is a widely used first-order algorithm for solving convex-concave saddle-point problems involving both smooth and nonsmooth components. Unlike the primal-dual hybrid gradient (PDHG) method, which incorporates an extrapolation step with parameter $\theta \in (0,1]$ to improve convergence, the existing convergence analysis of PAPC has been limited to the case … Read more

Lipschitz Based Lower Bound Construction for Surrogate Optimization

Published: 2024/01/30, Updated: 2025/04/20

Mohammadsina Almasi

Hadis Anahideh

Jay Rosenberger

Bound-constrained Optimization, Global Optimization, Global Optimization Applications lipschitz continuity, Lower Bound Construction, Nonlinear Distance Metrics, surrogate optimization

Bounds play a vital role in guiding optimization algorithms by enhancing convergence, improving solution quality, and quantifying optimality gaps. While Lipschitz-based lower bounds are well-established, their effectiveness is often constrained by the function’s topological properties. To address these limitations, we propose an approach that integrates nonlinear distance metrics with surrogate approximations, yielding more adaptive and … Read more

Convergence of Finite-Dimensional Approximations for Mixed-Integer Optimization with Differential Equations

Published: 2018/12/05, Updated: 2019/12/16

Falk M. Hante

Martin Schmidt

(Mixed) Integer Nonlinear Programming, Nonlinear Optimization, Systems governed by Differential Equations Optimization lipschitz continuity, mixed-integer nonlinear programming, optimal value function, optimization with differential equations, parametric optimization

We consider a direct approach to solve mixed-integer nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient conditions for the convergence of this approach … Read more

On Stable Piecewise Linearization and Generalized Algorithmic Differentiation

Published: 2012/09/05, Updated: 2012/12/30

Andreas Griewank

Nonlinear Optimization, Nonsmooth Optimization, Optimization Software Design Principles adol-c, automatic differentiation, bouligand derivative, bundle method, coherent orientation, computational graph, conical activity, directional derivative, generalized gradients and jacobians, lipschitz continuity, midpoint method, piecewise differentiability, piecewise newton, trapezoidal rule

It is shown how functions that are defined by evaluation programs involving the absolute value function (besides smooth elementals), can be approximated locally by piecewise-linear models in the style of algorithmic, or automatic, differentiation (AD). The model can be generated by a minor modification of standard AD tools and it is Lipschitz continuous with respect … Read more

On smooth relaxations of obstacle sets

Published: 2011/08/18, Updated: 2013/06/11

Oliver Stein

Paul Steuermann

Constrained Nonlinear Optimization, Convex Optimization entropic smoothing, error bound, hyperbolic smoothing, lipschitz continuity, obstacle problem, relaxation

We present and discuss a method to relax sets described by finitely many smooth convex inequality constraints by the level set of a single smooth convex inequality constraint. Based on error bounds and Lipschitz continuity, special attention is paid to the maximal approximation error and a guaranteed safety margin. Our results allow to safely avoid … Read more

Generalized differentiation with positively homogeneous maps: Applications in set-valued analysis and metric regularity

Published: 2010/02/16

C.H. Jeffrey Pang

Convex and Nonsmooth Optimization calmness, coderivatives, lipschitz continuity, metric regularity, multi-function differentiability

We propose a new concept of generalized differentiation of set-valued maps that captures the first order information. This concept encompasses the standard notions of Frechet differentiability, strict differentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between the Aubin … Read more

Fischer-Burmeister Complementarity Function on Euclidean Jordan Algebras

Published: 2007/12/15

Lingchen Kong

Levent Tunçel

Naihua Xiu

Complementarity and Variational Inequalities euclidean jordan algebra, fischer-burmeister function, lipschitz continuity

Recently, Gowda et al. [10] established the Fischer-Burmeister (FB) complementarity function (C-function) on Euclidean Jordan algebras. In this paper, we prove that FB C-function as well as the derivatives of the squared norm of FB C-function are Lipschitz continuous. CitationResearch Report CORR 2007-17, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, November … Read more