In Chapter 4 we have reviewed a parametric solution approach for multiobjective convex quadratic optimization problems using the weight space decomposition by efficient complementary bases of the parametric linear complementarity problem (pLCP). We have shown that there exists a one-to-one cor- respondence between efficient complementary bases and efficient active sets.
We proposed an algorithm for the determination of all efficient complementary bases without symbolic computations. We considered a generalization of multiobjective convex problems in canonical form to the general form.
Three special cases were discussed for which the weight cells are convex polyhedra. Multiobjective convex quadratic problems with diagonal objective matrices and lower and upper bounds were dis- cussed and it was shown that the weight space decomposition of such problems is an arrangement of hyperplanes. This provides a polynomial bound on the number of efficient complementary bases (for a fixed m). Furthermore, we have found a multiobjective linear programming problem with a weight space decomposition that has the same arrangement of hyperplanes.
Furthermore, we considered the parameter space decomposition for the e-constraint scalarization that can be computed using the weight space decomposition. An application of this relationship was used to provide a method for the computation of the e-constraint parameter space decomposition for mul- tiobjective location problems with l2and l2
2norms.
For multiobjective convex quadratic optimization problems with more than 3 objectives it is difficult to use the analytic description of the efficient set provided by the weight space decomposition, as the weight cells are in general m − 1-dimensional semi-algebraic sets [1].
An approach to approximate the weight cells is discussed in Chapter 5 using multiobjective convex piecewise-linear optimization problems.
An interesting question is whether the weight space decomposition by active sets can be generalized to other multiobjective convex optimization problems, such as multiobjective convex polynomial opti- mization problems.
Approximation of Multiobjective
Convex Optimization Problems by
Multiobjective Piecewise-Linear
Problems
Many multiobjective optimization problems have a large number of efficient solutions - in the case of continuous optimization the efficient set can be infinite.
From a theoretical or technical standpoint it may be difficult to compute a complete description of the efficient set, even if an analytical description is available. We observed in Chapter 4 that the efficient set of multiobjective convex programming problems can be described analytically using efficient active sets and parametric optimization. However, we also observed that, apart from some special cases, computing the analytical representation is difficult and in large-dimensional cases impractical. Furthermore, for real-world problems the aim is often to find an efficient solution that satisfies the preferences of the decision maker. For this task it may be sufficient to provide the decision maker with an approximation of the efficient set or the nondominated set, respectively.
For these reasons a variety of approaches have been provided in the literature that aim to compute a representation of the efficient set or the nondominated set, or both, for different types of multiobjective problems [74].
One category of such approaches are point approximations for which different quality measures are discussed in the literature, such as the Hausdorff distance between the nondominated set and the representation set (referred to as coverage) and uniformity [77, 78]. Point approximations can be computed using different techniques, for example dichotomic search [23, 71, 72] or evolutionary al- gorithms [10, 83].
A number of approaches construct a piecewise-linear inner or outer approximation of the nondomi- nated set of convex multiobjective optimization problems [7, 25, 26, 52, 53]. We refer to [74] for a detailed survey.
In this chapter, we will consider an approach proposed by Oberdieck and Pistikopoulos [67] for approx- imating the weight space decomposition of multiobjective convex quadratic optimization problems. In this approach the objective functions are approximated by piecewise-linear functions. A weight space decomposition of the multiobjective piecewise-linear problem can then be computed.
In Section 5.1 the weight space decomposition for multiobjective convex continuous optimization prob- lems is introduced.
In Section 5.2 multiobjective convex piecewise-linear programming problems and the corresponding formulation as a multiobjective linear programming problem are reviewed. Additionally, the weight space decomposition for multiobjective convex piecewise-linear programming problems is introduced.
The properties of the weight space decomposition for different types of multiobjective convex opti- mization problems are summarized in Section 5.3 using results from Chapter 4 and the literature [23]. In Section 5.4 the outer approximation approach by Oberdieck and Pistikopoulos [67] is reviewed. We discuss the convergence properties using results from the field of approximation of convex compact sets by polyhedrons.
In Section 5.5 a we construct an approximation of the weight space decomposition of a triobjective convex quadratic optimization problem and discuss the result.
5.1
A Weight Space Decomposition for Multiobjective Convex Op-
timization Problems
In this section we will first discuss an extension of the weight space decomposition by active sets for multiobjective convex quadratic optimization problems.
Consider a multiobjective convex programming problem with linear equality and inequality constraints: vmin
x∈Rn fi(x) i = 1, . . . , m
s.t. Ax ½ b, xI+½ 0, H x= h (MCP) with convex objective functions fi(x) for i = 1, . . . , n, A ∈ Rp×n, b ∈ Rp, H ∈ Rq×n, h ∈ Rq and index set I+⊆ {1, . . . , n} of nonnegative variables.
The feasible set is denoted by
S = x ∈ Rn : Ax ½ b, H x = h, x
I+½ 0
.
Additionally, we assume that the objective functions fi(x), i = 1, . . . , m, are continuous, but not neces- sarily differentiable, on S.
The weighted sum scalarization problem of (MCP) for λ ∈ Λ is given by min x∈Rn Pm i=1λifi(x) s.t. Ax ½ b, xI+½ 0, H x= h (WCP)
The set of optimal solutions of the weighted sum scalarization problem (WCP) is denoted by Xopt(λ) for a fixed λ ∈ Λ.
In Definition 4.52 the efficient active sets for multiobjective convex quadratic optimization problems (gMQP) are defined using the KKT conditions. Since we do not assume that the objective functions of (MCP) fi(x) are differentiable for all i = 1, . . . , m we consider here the following definition of active sets:
Definition 5.1. Let XE be the efficient set of (MCP). The active set A (x) = (I(x), J(x)) with I⊂ I+of a feasible point x ∈ S is defined as:
I(x) =
j ∈ I+ : xj= 0 and J(x) = j ∈ {1, . . . , p} : Ajx = bj . (5.1) An active set ¯A = (¯I, ¯J) is called efficient active set of (MCP) if there exists x ∈ XE such that A (x) = ¯A .
The set of efficient active sets is defined as:
Aeff= {A (x) : x ∈ XE} (5.2)
For a given efficient active set ¯A we can define the weights λ ∈ Λ such that there exists an optimal solution x ∈ S of the weighted sum scalarization problem (WCP) with A (x) = ¯A :
ΛC( ¯A ) :=λ ∈ Λ : ∃x ∈ Xopt(λ) ∩ XE such that A (x) = ¯A (5.3) In general, it is difficult to determine the set ΛC
(A ) for multiobjective convex problems for which no analytical or closed-formula solution of the weighted sum scalarization problem is available.
Two classes of multiobjective convex optimization problems have been discussed in this dissertation for which the weight cells can be determined analytically: Multiobjective linear programming problems, which were introduced in Section 2.2.3, and multiobjective convex quadratic optimization problems, which were discussed in Chapter 4.
In Section 5.2 we will discuss another class of multiobjective convex optimization problems for which a decomposition of the weight space can be computed explicitly.