arXiv subjects

Federico Battista

Publications and source records attributed to Federico Battista.

3 recordsLinked to original sources

arXiv2025

Improving Directions in Mixed Integer Bilevel Linear Optimization

We consider the central role of improving directions in solution methods for mixed integer bilevel linear optimization problems (MIBLPs). Current state-of-the-art methods for solving MIBLPs employ the branch-and-cut framework originally developed for solving mixed integer linear optimization problems. This approach relies on oracles for two kinds of subproblems: those for checking whether a candidate pair of leader's and follower's decisions is bilevel feasible, and those required for generating valid inequalities. Typically, these two types of oracles are managed separately, but in this work, we explore their close connection and propose a solution framework based on solving a single type of subproblem: determining whether there exists a so-called improving feasible direction for the follower's problem. Solution of this subproblem yields information that can be used both to check feasibility and to generate strong valid inequalities. Building on prior works, we expose the foundational role of improving directions in enforcing the follower's optimality condition and extend a previously known hierarchy of optimality-based relaxations to the mixed-integer setting, showing that the associated relaxed feasible regions coincide exactly with the closure associated with intersection cuts derived from improving directions. Numerical results with an implementation using a modified version of the open source solver MibS show that this approach can yield practical improvements.

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arXiv2025

Application of the Lovász-Schrijver Lift-and-Project Operator to Compact Stable Set Integer Programs

The Lovász theta function $θ(G)$ provides a very good upper bound on the stability number of a graph $G$. It can be computed in polynomial time by solving a semidefinite program (SDP), which also turns out to be fairly tractable in practice. Consequently, $θ(G)$ achieves a hard-to-beat trade-off between computational effort and strength of the bound. Indeed, several attempts to improve the theta bound are documented, mainly based on playing around the application of the $N_+(\cdot)$ lifting operator of Lovász and Schrijver to the classical formulation of the maximum stable set problem. Experience shows that solving such SDP-s often struggles against practical intractability and requires highly specialized methods. We investigate the application of such an operator to two different linear formulations based on clique and nodal inequalities, respectively. Fewer inequalities describe these two and yet guarantee that the resulting SDP bound is at least as strong as $θ(G)$. Our computational experience, including larger graphs than those previously documented, shows that upper bounds stronger than $θ(G)$ can be accessed by a reasonable additional effort using the clique-based formulation on sparse graphs and the nodal-based one on dense graphs.

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arXiv2023

Dealing with inequality constraints in large-scale semidefinite relaxations for graph coloring and maximum clique problems

Semidefinite programs (SDPs) can be solved in polynomial time by interior point methods. However, when the dimension of the problem gets large, interior point methods become impractical in terms of both computational time and memory requirements. Certain first-order methods, such as Alternating Direction Methods of Multipliers (ADMMs), established as suitable algorithms to deal with large-scale SDPs and gained growing attention over the past decade. In this paper, we focus on an ADMM designed for SDPs in standard form and extend it to deal with inequalities when solving SDPs in general form. Beside numerical results on randomly generated instances, where we show that our method compares favorably with respect to the state-of-the-art solver SDPNAL+, we present results on instances from SDP relaxations of classical combinatorial problems such as the graph coloring problem and the maximum clique problem. Through extensive numerical experiments, we show that even an inaccurate dual solution, obtained at a generic iteration of our proposed ADMM, can represent an efficiently recovered valid bound on the optimal solution of the combinatorial problems considered, as long as an appropriate post-processing procedure is applied.

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