1,284 research outputs found
Optimising a nonlinear utility function in multi-objective integer programming
In this paper we develop an algorithm to optimise a nonlinear utility
function of multiple objectives over the integer efficient set. Our approach is
based on identifying and updating bounds on the individual objectives as well
as the optimal utility value. This is done using already known solutions,
linear programming relaxations, utility function inversion, and integer
programming. We develop a general optimisation algorithm for use with k
objectives, and we illustrate our approach using a tri-objective integer
programming problem.Comment: 11 pages, 2 tables; v3: minor revisions, to appear in Journal of
Global Optimizatio
On the representation of the search region in multi-objective optimization
Given a finite set of feasible points of a multi-objective optimization
(MOO) problem, the search region corresponds to the part of the objective space
containing all the points that are not dominated by any point of , i.e. the
part of the objective space which may contain further nondominated points. In
this paper, we consider a representation of the search region by a set of tight
local upper bounds (in the minimization case) that can be derived from the
points of . Local upper bounds play an important role in methods for
generating or approximating the nondominated set of an MOO problem, yet few
works in the field of MOO address their efficient incremental determination. We
relate this issue to the state of the art in computational geometry and provide
several equivalent definitions of local upper bounds that are meaningful in
MOO. We discuss the complexity of this representation in arbitrary dimension,
which yields an improved upper bound on the number of solver calls in
epsilon-constraint-like methods to generate the nondominated set of a discrete
MOO problem. We analyze and enhance a first incremental approach which operates
by eliminating redundancies among local upper bounds. We also study some
properties of local upper bounds, especially concerning the issue of redundant
local upper bounds, that give rise to a new incremental approach which avoids
such redundancies. Finally, the complexities of the incremental approaches are
compared from the theoretical and empirical points of view.Comment: 27 pages, to appear in European Journal of Operational Researc
Optimizing over the Efficient Set of a Multi-Objective Discrete Optimization Problem
Optimizing over the efficient set of a discrete multi-objective problem is a challenging issue. The main reason is that, unlike when optimizing over the feasible set, the efficient set is implicitly characterized. Therefore, methods designed for this purpose iteratively generate efficient solutions by solving appropriate single-objective problems. However, the number of efficient solutions can be quite large and the problems to be solved can be difficult practically. Thus, the challenge is both to minimize the number of iterations and to reduce the difficulty of the problems to be solved at each iteration.
In this paper, a new enumeration scheme is proposed. By introducing some constraints and optimizing over projections of the search region, potentially large parts of the search space can be discarded, drastically reducing the number of iterations. Moreover, the single-objective programs to be solved can be guaranteed to be feasible, and a starting solution can be provided allowing warm start resolutions. This results in a fast algorithm that is simple to implement.
Experimental computations on two standard multi-objective instance families show that our approach seems to perform significantly faster than the state of the art algorithm
Relaxations and Duality for Multiobjective Integer Programming
Multiobjective integer programs (MOIPs) simultaneously optimize multiple
objective functions over a set of linear constraints and integer variables. In
this paper, we present continuous, convex hull and Lagrangian relaxations for
MOIPs and examine the relationship among them. The convex hull relaxation is
tight at supported solutions, i.e., those that can be derived via a
weighted-sum scalarization of the MOIP. At unsupported solutions, the convex
hull relaxation is not tight and a Lagrangian relaxation may provide a tighter
bound. Using the Lagrangian relaxation, we define a Lagrangian dual of an MOIP
that satisfies weak duality and is strong at supported solutions under certain
conditions on the primal feasible region. We include a numerical experiment to
illustrate that bound sets obtained via Lagrangian duality may yield tighter
bounds than those from a convex hull relaxation. Subsequently, we generalize
the integer programming value function to MOIPs and use its properties to
motivate a set-valued superadditive dual that is strong at supported solutions.
We also define a simpler vector-valued superadditive dual that exhibits weak
duality but is strongly dual if and only if the primal has a unique
nondominated point
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