A Proximal Gradient Method for Multi-objective Optimization Problems Using Bregman Functions

In this paper, a globally convergent proximal gradient method is developed for convex multi-objective optimization problems using Bregman distance. The proposed method is free from any kind of a priori chosen parameters or ordering information of objective functions. At every iteration of the proposed method, a subproblem is solved to find a descent direction. This … Read more

Advancements in the computation of enclosures for multi-objective optimization problems

A central goal for multi-objective optimization problems is to compute their nondominated sets. In most cases these sets consist of infinitely many points and it is not a practical approach to compute them exactly. One solution to overcome this problem is to compute an enclosure, a special kind of coverage, of the nondominated set. One … Read more

Convergence rates of the stochastic alternating algorithm for bi-objective optimization

Stochastic alternating algorithms for bi-objective optimization are considered when optimizing two conflicting functions for which optimization steps have to be applied separately for each function. Such algorithms consist of applying a certain number of steps of gradient or subgradient descent on each single objective at each iteration. In this paper, we show that stochastic alternating … Read more

Pattern-based ILP models for the one-dimensional cutting stock problem with setup cost

The One-Dimensional Cutting Stock Problem with Setup Cost (CSP-S) is a cutting problem that seeks a cutting plan with a minimum number of objects and a minimum number of different patterns. This problem gains relevance in manufacturing settings, where time consuming operations to set up the knives of the cutting machine for the new patterns … Read more

A hybrid patch decomposition approach to compute an enclosure for multi-objective mixed-integer convex optimization problems

In multi-objective mixed-integer convex optimization multiple convex objective functions need to be optimized simultaneously while some of the variables are only allowed to take integer values. In this paper we present a new algorithm to compute an enclosure of the nondominated set of such optimization problems. More precisely, we decompose the multi-objective mixed-integer convex optimization … Read more

On implementation details and numerical experiments for the HyPaD algorithm to solve multi-objective mixed-integer convex optimization problems

In this paper we present insights on the implementation details of the hybrid patch decomposition algorithm (HyPaD) for convex multi-objective mixed-integer optimization problems. We discuss how to implement the SNIA procedure which is basically a black box algorithm in the original work by Eichfelder and Warnow. In addition, we present and discuss results for various … Read more

An Upper Bound on the Hausdorff Distance Between a Pareto Set and its Discretization in Bi-Objective Convex Quadratic Optimization

We provide upper bounds on the Hausdorff distances between the efficient set and its discretization in the decision space, and between the Pareto set (also called the Pareto front) and its discretization in the objective space, in the context of bi-objective convex quadratic optimization on a compact feasible set. Our results imply that if t … Read more

An approximation algorithm for multi-objective optimization problems using a box-coverage

For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the … Read more

Accuracy and fairness trade-offs in machine learning: A stochastic multi-objective approach

In the application of machine learning to real life decision-making systems, e.g., credit scoring and criminal justice, the prediction outcomes might discriminate against people with sensitive attributes, leading to unfairness. The commonly used strategy in fair machine learning is to include fairness as a constraint or a penalization term in the minimization of the prediction … Read more

The Magic of Nash Social Welfare in Optimization: Do Not Sum, Just Multiply!

In this paper, we explain some key challenges when dealing with a single/multi-objective optimization problem in practice. To overcome these challenges, we present a mathematical program that optimizes a Nash Social Welfare function. We refer to this mathematical program as the Nash Social Welfare Program (NSWP). An interesting property of the NSWP is that it … Read more