The Birkhoff polytope (the convex hull of the set of permutation matrices), which is represented using $\Theta(n^2)$ variables and constraints, is frequently invoked in formulating relaxations of optimization problems over permutations. Using a recent construction of Goemans (2010), we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of variables and constraints to $\Theta(n \log n)$ in theory and $\Theta(n \log^2 n)$ in practice. We modify the recent convex formulation of the 2-SUM problem introduced by Fogel et al. (2013) to use this polytope, and demonstrate how we can attain results of similar quality in significantly less computational time for large $n$. To our knowledge, this is the first usage of Goemans' compact formulation of the permutahedron in a convex optimization problem. We also introduce a simpler regularization scheme for this convex formulation of the 2-SUM problem that yields good empirical results.

A recent trend in the mathematical optimization literature is to embed trained machine learning predictors into a larger optimization model. The m ost common application is for a practitioner to train a machine learning predictor as a sur rogate for a more complicated subsystem that cannot be directly embedded into an optimiza tion model, for example, because it does not have an algebraic form or because it is non -differentiable. L opez-Flores et al. (2024) provide a review of the field.

The Birkhoff polytope (the convex hull of the set of permutation matrices), which is represented using \Theta(n 2) variables and constraints, is frequently invoked in formulating relaxations of optimization problems over permutations. Using a recent construction of Goemans (2010), we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of variables and constraints to \Theta(n \log n) in theory and \Theta(n \log 2 n) in practice. We modify the recent convex formulation of the 2-SUM problem introduced by Fogel et al. (2013) to use this polytope, and demonstrate how we can attain results of similar quality in significantly less computational time for large n . To our knowledge, this is the first usage of Goemans' compact formulation of the permutahedron in a convex optimization problem. We also introduce a simpler regularization scheme for this convex formulation of the 2-SUM problem that yields good empirical results.

A factored Nonlinear Program (Factored-NLP) explicitly models the dependencies between a set of continuous variables and nonlinear constraints, providing an expressive formulation for relevant robotics problems such as manipulation planning or simultaneous localization and mapping. When the problem is over-constrained or infeasible, a fundamental issue is to detect a minimal subset of variables and constraints that are infeasible. Previous approaches require solving several nonlinear programs, incrementally adding and removing constraints, and are thus computationally expensive. In this paper, we propose a graph neural architecture that predicts which variables and constraints are jointly infeasible. The model is trained with a dataset of labeled subgraphs of Factored-NLPs, and importantly, can make useful predictions on larger factored nonlinear programs than the ones seen during training. We evaluate our approach in robotic manipulation planning, where our model is able to generalize to longer manipulation sequences involving more objects and robots, and different geometric environments. The experiments show that the learned model accelerates general algorithms for conflict extraction (by a factor of 50) and heuristic algorithms that exploit expert knowledge (by a factor of 4).

My career as a practitioner and researcher in the data science space has spanned more than 30 years, and during that time I have seen a lot of new advanced analytics technologies – which were touted as "the latest and greatest," "cutting-edge," or "game-changing" or another similar superlative – sizzle and then fizzle. The hype cycles (as Gartner calls them) of these technologies were short – as they failed to deliver real-world business impact and attain long-term commercial viability. One advanced analytics technology that bucks that trend and has been around ever since I entered the professional arena in the early 1990s (and actually long before that with the introduction of linear programming in the 1940s) is mathematical optimization. For decades, mathematical optimization has been widely used by companies of all sizes and stripes to address their complex business problems. The secret to mathematical optimization's staying power is that it has consistently demonstrated that it is capable of generating optimal solutions to large-scale, real-world business problems – and has thereby produced significant business value.

This paper addresses a variant of multi-agent path finding (MAPF) in continuous space and time. We present a new solving approach based on satisfiability modulo theories (SMT) to obtain makespan optimal solutions. The standard MAPF is a task of navigating agents in an undirected graph from given starting vertices to given goal vertices so that agents do not collide with each other in vertices of the graph. In the continuous version (MAPF-R) agents move in an n-dimensional Euclidean space along straight lines that interconnect predefined positions. For simplicity, we work with circular omni-directional agents having constant velocities in the 2D plane.

Scheduling and facility location represent two classes of well-studied combinatorial optimization problems. The main motivation for studying them relies on the broad range of applications (e.g., in public services, industry, logistics, project management, production planning, data processing, etc.), as well as on the challenge in providing efficient solutions, since many of these problems are classified as NPhard (see, e.g., Pinedo 2009, Pinedo 2016, Drezner and Hamacher 2002, and Laporte et al. 2015). Since the 1960s, many works on these topics have been published, but only a few of them has focused on studying these problems in an integrated fashion. Due to the limited capacity of the computers of two decades ago, it was usual to solve integrated combinatorial optimization problems using sequential approaches, i.e., solving each problem separately in such a way that the solution of one represents an input to the other. However, this strategy does not guarantee the optimality of the overall solution and, in addition, the input solutions may not be feasible for the successor problems. With the recent advances in technology, especially in the computational field, solving integrated combinatorial optimization problems using integrated approaches is becoming more accessible. In this context, the ScheLoc problem combines the job scheduling and facility location in a single and integrated problem.

The Birkhoff polytope (the convex hull of the set of permutation matrices), which is represented using $\Theta(n 2)$ variables and constraints, is frequently invoked in formulating relaxations of optimization problems over permutations. Using a recent construction of Goemans (2010), we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of variables and constraints to $\Theta(n \log n)$ in theory and $\Theta(n \log 2 n)$ in practice. We modify the recent convex formulation of the 2-SUM problem introduced by Fogel et al. (2013) to use this polytope, and demonstrate how we can attain results of similar quality in significantly less computational time for large $n$. To our knowledge, this is the first usage of Goemans' compact formulation of the permutahedron in a convex optimization problem. We also introduce a simpler regularization scheme for this convex formulation of the 2-SUM problem that yields good empirical results.

Abstract--Backtracking search algorithms are often used to solve the Constraint Satisfaction Problem (CSP). The efficiency of backtracking search depends greatly on the variable ordering heuristics. Currently, the most commonly used heuristics are handcrafted based on expert knowledge. In this paper, we propose a deep reinforcement learning based approach to automatically discover new variable ordering heuristics that are better adapted for a given class of CSP instances. We show that directly optimizing the search cost is hard for bootstrapping, and propose to optimize the expected cost of reaching a leaf node in the search tree. T o capture the complex relations among the variables and constraints, we design a representation scheme based on Graph Neural Network that can process CSP instances with different sizes and constraint arities. Experimental results on random CSP instances show that the learned policies outperform classical handcrafted heuristics in terms of minimizing the search tree size, and can effectively generalize to instances that are larger than those used in training. Constraint Satisfaction Problem (CSP) is one of the most widely studied problems in computer science and artificial intelligence. It provides a common framework for modeling and solving combinatorial problems in many application domains, such as planning and scheduling [1], [2], vehicle routing [3], [4], graph problems [5], [6], and computational biology [7], [8]. A CSP instance involves a set of variables and constraints. T o solve it, one needs to find a value assignment for all variables such that all constraints are satisfied, or prove such assignment does not exist. Despite its ubiquitous applications, unfortunately, CSP is well known to be NPcomplete in general [9]. T o solve CSP efficiently, backtracking search algorithms are often employed, which are exact algorithms with the guarantee that a solution will be found if one exists.