An objective space based algorithm for biobjective mixed integer programming problems
Firdevs Ulus, Ozlem Karsu, Deniz Emre
We propose an objective space based exact solution algorithm for biobjective mixed integer programming problems. The algorithm solves scalarization models in order to explore predetermined regions of the objective space called boxes, defined by two nondominated points. At each iteration of the algorithm, a box is explored either by a weighted sum or a Pascoletti-Serafini scalarization to determine nondominated line segments and points. We demonstrate the applicability of the algorithm through computational experiments.
Toward a Branch-and-Bound Algorithm for Biobjective Mixed Integer Quadratic Programs
Biobjective quadratic programs with mixed-integer variables (BOMIQPs) model decision making situations encountered in finance, economics, engineering, and other areas of human activity. We present the work we have accomplished so far toward development of a branch and bound (BB) algorithm for convex BOMIQPs. The algorithm consists of the following components: (i) algorithms for solving auxiliary single objective parametric quadratic programs; (ii) fathoming rules; (iii) branching rules; (iv) rules to establish (non)dominance between sets; (v) rules for updating the incumbent nondominated set.
Finding nondominated and efficient solutions of multiobjective integer quadratic programming problems
Marianna De Santis, Gabriele Eichfelder
We present a deterministic method for minimizing multiple quadratic objective functions over integer variables. Our method looks for efficient points by fixing subsets of variables to integer values and by using lower bounds in the form of hyperplanes in the image space derived from the continuous relaxations of the restricted objective functions. We show that the algorithm finds all efficient and all nondominated points after finitely many fixings of variables. Numerical examples on bi-objective instances as well as on instances with three and four objectives are shown.
Pareto Front Approximation through a Multi-objective Augmented Lagrangian Method
Pierluigi Mansueto, Matteo Lapucci, Guido Cocchi
In this work, we propose an extension of a multi-objective augmented Lagrangian Method from recent literature suitable for smooth convex constrained multi-objective optimization problems. The new algorithm is designed to handle sets of points and produce good Pareto front approximations, as opposed to the original one which converges to a single solution. We prove properties of Pareto stationarity global convergence for the sequences of points generated by our method. We then compare it with main state-of-the-art algorithms on some problems, showing its effectiveness and general superiority.