A Parametric Approach for Solving Convex Quadratic Optimization with Indicators Over Trees

This paper investigates convex quadratic optimization problems involving $n$ indicator variables, each associated with a continuous variable, particularly focusing on scenarios where the matrix $Q$ defining the quadratic term is positive definite and its sparsity pattern corresponds to the adjacency matrix of a tree graph. We introduce a graph-based dynamic programming algorithm that solves this … Read more

A Proximal-Gradient Method for Constrained Optimization

We present a new algorithm for solving optimization problems with objective functions that are the sum of a smooth function and a (potentially) nonsmooth regularization function, and nonlinear equality constraints. The algorithm may be viewed as an extension of the well-known proximal-gradient method that is applicable when constraints are not present. To account for nonlinear … Read more

Strengthening Lasserre’s Hierarchy in Real and Complex Polynomial Optimization

We establish a connection between multiplication operators and shift operators. Moreover, we derive positive semidefinite conditions of finite rank moment sequences and use these conditions to strengthen Lasserre’s hierarchy for real and complex polynomial optimization. Integration of the strengthening technique with sparsity is considered. Extensive numerical experiments show that our strengthening technique can significantly improve … Read more

Tricks from the Trade for Large-Scale Markdown Pricing: Heuristic Cut Generation for Lagrangian Decomposition

In automated decision making processes in the online fashion industry, the ‘predict-then-optimize’ paradigm is frequently applied, particularly for markdown pricing strategies. This typically involves a mixed-integer optimization step, which is crucial for maximizing profit and merchandise volume. In practice, the size and complexity of the optimization problem is prohibitive for using off-the-shelf solvers for mixed … Read more

A Sequential Benders-based Mixed-Integer Quadratic Programming Algorithm

For continuous decision spaces, nonlinear programs (NLPs) can be efficiently solved via sequential quadratic programming (SQP) and, more generally, sequential convex programming (SCP). These algorithms linearize only the nonlinear equality constraints and keep the outer convex structure of the problem intact, such as (conic) inequality constraints or convex objective terms. The aim of the presented … Read more

A Max-Min-Max Algorithm for Large-Scale Robust Optimization

Robust optimization (RO) is a powerful paradigm for decision making under uncertainty. Existing algorithms for solving RO, including the reformulation approach and the cutting-plane method, do not scale well, hindering the application of RO to large-scale decision problems. In this paper, we devise a first-order algorithm for solving RO based on a novel max-min-max perspective. … Read more

Revisiting the fitting of the Nelson-Siegel and Svensson models

The Nelson-Siegel and the Svensson models are two of the most widely used models for the term structure of interest rates. Even though the models are quite simple and intuitive, fitting them to market data is numerically challenging and various difficulties have been reported. In this paper, a novel mathematical analysis of the fitting problem … Read more

Polyhedral Analysis of Quadratic Optimization Problems with Stieltjes Matrices and Indicators

In this paper, we consider convex quadratic optimization problems with indicators on the continuous variables. In particular, we assume that the Hessian of the quadratic term is a Stieltjes matrix, which naturally appears in sparse graphical inference problems and others. We describe an explicit convex formulation for the problem by studying the Stieltjes polyhedron arising … Read more

The best approximation pair problem relative to two subsets in a normed space

In the classical best approximation pair (BAP) problem, one is given two nonempty, closed, convex and disjoint subsets in a finite- or an infinite-dimensional Hilbert space, and the goal is to find a pair of points, each from each subset, which realizes the distance between the subsets. This problem, which has a long history, has … Read more

An MILP-Based Solution Scheme for Factored and Robust Factored Markov Decision Processes

Factored Markov decision processes (MDPs) are a prominent paradigm within the artificial intelligence community for modeling and solving large-scale MDPs whose rewards and dynamics decompose into smaller, loosely interacting components. Through the use of dynamic Bayesian networks and context-specific independence, factored MDPs can achieve an exponential reduction in the state space of an MDP and … Read more