Global Approaches for Facility Layout and VLSI Floorplanning

This paper summarizes recent advances in the global solution of several relevant facility layout problems

Global Approaches for Facility Layout and VLSI Floorplanning

This paper summarizes recent advances in the global solution of several relevant facility layout problems

A New Mathematical Programming Framework for Facility Layout Design

We present a new framework for efficiently finding competitive solutions for the facility layout problem. This framework is based on the combination of two new mathematical programming models. The first model is a relaxation of the layout problem and is intended to find good starting points for the iterative algorithm used to solve the second model. The second model is an exact formulation of the facility layout problem as a non-convex mathematical program with equilibrium constraints (MPEC). Aspect ratio constraints, which are frequently used in facility layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts, are easily incorporated into this new framework. Finally, we present computational results showing that both models, and hence the complete framework, can be solved efficiently using widely available optimization software. This important feature of the new framework implies that it can be used to find competitive layouts with relatively little computational effort. This is advantageous for a user who wishes to consider several competitive layouts rather than simply using the mathematically optimal layout

A class of spectral bounds for Max k-Cut

International audienceIn this paper we introduce a new class of bounds for the maximum -cut problem on undirected edge-weighted simple graphs. The bounds involve eigenvalues of the weighted adjacency matrix together with geometrical parameters. They generalize previous results on the maximum (2-)cut problem and we demonstrate that they can strictly improve over other eigenvalue bounds from the literature. We also report computational results illustrating the potential impact of the new bounds

Learning Deterministic Surrogates for Robust Convex QCQPs

Decision-focused learning is a promising development for contextual optimisation. It enables us to train prediction models that reflect the contextual sensitivity structure of the problem. However, there have been limited attempts to extend this paradigm to robust optimisation. We propose a double implicit layer model for training prediction models with respect to robust decision loss in uncertain convex quadratically constrained quadratic programs (QCQP). The first layer solves a deterministic version of the problem, the second layer evaluates the worst case realisation for an uncertainty set centred on the observation given the decisions obtained from the first layer. This enables us to learn model parameterisations that lead to robust decisions while only solving a simpler deterministic problem at test time. Additionally, instead of having to solve a robust counterpart we solve two smaller and potentially easier problems in training. The second layer (worst case problem) can be seen as a regularisation approach for predict-and-optimise by fitting to a neighbourhood of problems instead of just a point observation. We motivate relaxations of the worst-case problem in cases of uncertainty sets that would otherwise lead to trust region problems, and leverage various relaxations to deal with uncertain constraints. Both layers are typically strictly convex in this problem setting and thus have meaningful gradients almost everywhere. We demonstrate an application of this model on simulated experiments. The method is an effective regularisation tool for decision-focused learning for uncertain convex QCQPs.Comment: Under submission at CPAIOR 202

Mathematical Optimization Approach for Facility Layout on Several Rows

The facility layout problem is concerned with finding an arrangement of non-overlapping indivisible departments within a facility so as to minimize the total expected flow cost. In this paper we consider the special case of multi-row layout in which all the departments are to be placed in three or more rows, and our focus is on, for the first time, solutions for large instances. We first propose a new mixed integer linear programming formulation that uses continuous variables to represent the departments’ location in both x and y coordinates, where x represents the position of a department within a row and y represents the row assigned to the department. We prove that this formulation always achieves an optimal solution with integer values of y, but it is limited to solving instances with up to 13 departments. This limitation motivates the application of a two-stage optimization algorithm that combines two mathematical optimization models by taking the output of the first-stage model as the input of the second-stage model. This algorithm is, to the best of our knowledge, the first one in the literature reporting solutions for instances with up to 100 departments.publishersversionpublishe

A Branch-and-Cut Algorithm based on Semidefinite Programming for the Minimum k-Partition Problem

The minimum k-partition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) for the MkP problem. The two key ingredients for this algorithm are: the combination of semidefinite programming (SDP) with polyhedral results; and the iterative clustering heuristic (ICH) that finds feasible solutions for the MkP problem. We compare ICH to the hyperplane rounding techniques of Goemans and Williamson and of Frieze and Jerrum, and the computational results support the conclusion that ICH consistently provides better feasible solutions for the MkP problem. ICH is used in our SBC algorithm to provide feasible solutions at each node of the branch-and-bound tree. The SBC algorithm computes globally optimal solutions for dense graphs with up to 60 vertices, for grid graphs with up to 100 vertices, and for different values of k, providing the best exact approach to date for k > 2