Repository – Page 221 – Optimization Online

An (s^r)hBcResolution ODE Framework for Understanding Discrete-Time Algorithms and Applications to the Linear Convergence of Minimax Problems

Published: 2021/01/02

There has been a long history of using ordinary differential equations (ODEs) to understand the dynamic of discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to obtain a \emph{suitable} ODE from a given DTA, and (ii) it is unclear the connection between the convergence of a … Read more

A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization

Published: 2021/01/01, Updated: 2021/04/13

(Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming applications, bilevel optimization, Branch-and-Bound, branch-and-cut, mixed-integer programming, survey

Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and … Read more

Optimization with learning-informed differential equation constraints and its applications

Published: 2021/01/01

Guozhi Dong

Michael Hintermueller

Kostas Papafitsoros

Biomedical Applications, Infinite Dimensional Optimization, Nonlinear Optimization

Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis … Read more

Constrained and Composite Optimization via Adaptive Sampling Methods

Published: 2020/12/30

Convex and Nonsmooth Optimization, Nonlinear Optimization, Stochastic Programming

The motivation for this paper stems from the desire to develop an adaptive sampling method for solving constrained optimization problems in which the objective function is stochastic and the constraints are deterministic. The method proposed in this paper is a proximal gradient method that can also be applied to the composite optimization problem min f(x) … Read more

Supermodularity and valid inequalities for quadratic optimization with indicators

Published: 2020/12/29

Alper Atamturk

Andrés Gómez

(Mixed) Integer Nonlinear Programming, Quadratic Programming conic quadratic cuts, convex piecewise valid inequalities, lifting, perspective formulation, quadratic programming, supermodular inequalities

We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the … Read more

Weak notions of nondegeneracy in nonlinear semidefinite programming

Published: 2020/12/29, Updated: 2022/03/15

Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming, Semi-definite Programming constraint nondegeneracy, constraint qualifications, semidefinite programming

The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, $\ell$-dimensional kernel of the constraint matrix, by the linear independence of a set of $\ell(\ell+1)/2$ derivative vectors. We show that this … Read more

Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications

Published: 2020/12/29

Fatma Kılınç-Karzan

Simge Küçükyavuz

Dabeen Lee

(Mixed) Integer Nonlinear Programming best subset selection, conic mixed-binary sets, conic quadratic optimization, convex hull, distributionally robust optimization, extended polymatroid inequalities, fractional binary optimization, submodularity

We consider a general conic mixed-binary set where each homogeneous conic constraint involves an affine function of independent continuous variables and an epigraph variable associated with a nonnegative function, $f_j$, of common binary variables. Sets of this form naturally arise as substructures in a number of applications including mean-risk optimization, chance-constrained problems, portfolio optimization, lot-sizing … Read more

Fleet Sizing and Service Region Partitioning for Same-Day Delivery Systems

Published: 2020/12/28, Updated: 2021/09/20

Dipayan Banerjee

Alan L. Erera

Alejandro Toriello

Applications - OR and Management Sciences, Transportation fleet sizing, partitioning, same-day delivery, tactical design

We study the linked tactical design problems of fleet sizing and partitioning a service region into vehicle routing zones for same-day delivery (SDD) systems. Existing SDD studies focus primarily on operational dispatch problems and do not consider system design questions. Prior work on SDD system design has not considered the fleet sizing decision when a … Read more

The Landscape of the Proximal Point Method for Nonconvex-Nonconcave Minimax Optimization

Published: 2020/12/27

Robust Optimization, Unconstrained Optimization minimax optimization, moreau envelope, nonconcave, nonconvex, nonsmooth, proximal point algorithm, smoothing

Minimax optimization has become a central tool for modern machine learning with applications in generative adversarial networks, robust optimization, reinforcement learning, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify and deal with the fundamental difficulties posed by nonconvex-nonconcave structures. In this paper, we study the classic proximal point method … Read more