Interactive Polyhedral Outer Approximation (IPOA) strategy for general multiobjective optimization problems

Abstract

We propose an interactive polyhedral outer approximation (IPOA) method to solve a broad class of multiobjective optimization problems (MOP) with, possibly, nonlinear and nondifferentiable objective and constraint functions, and with continuous or discrete decision variables. During the interactive optimization phase, the method progressively constructs a polyhedral approximation of the decision-maker’s (DM’s) unknown preference structure and a polyhedral outer-approximation of the feasible set of MOP. The piecewise linear approximation of the DM’s preferences also provides a mechanism for testing the consistency of the DM’s assessments and removing inconsistencies; it also allows post-optimality analysis. All the feasible trial solutions are non-dominated (efficient, or Pareto-optimal) so preference assessments are made in the context of non-dominated alternatives only. Upper and lower bounds on the yet unknown optimal value are produced at every iteration, allowing terminating the search prematurely at a good-enough solution and providing information about the closeness of this solution to the optimal solution. The IPOA method includes a preliminary phase in which a limited probe of the efficient set is conducted in order to find a good initial trial solution for the interactive phase. The computational requirements of the algorithm are relatively simple. The results of an extensive computational study are reported.

Appendices

Appendix A: Proofs

Proof of Theorem 4.1

Assume that text equation (15) holds, but that x ⁱ∉Ω. Then g _j (x ⁱ)>0 for one or more j. Take (without loss of generality) x=x ⁱ, ξ _j >0 for j such that g _j (x ⁱ)>0 and ξ _j =0 otherwise, and let ξ=(ξ ₁,…,ξ _p ), π=(π ₁,…,π _n ) where $\pi\in \sum^{p}_{j=1} \xi_{j} \partial _{x} g_{j}(x)$. Then ξg(x)+π(x ⁱ−x)=ξg(x)>0, which contradicts the assumption that (15) holds for all x∈X ^∗, ξ≥0. Hence, x ⁱ∈Ω. □

Proof of Theorem 4.2

The inequality $\theta^{k} \le u_{k}^{k}$, or θ ^k≤u ^k(F(x ^k)) (upon using $u^{k}(F(x^{k})) = u_{k}^{k}$: see text equation (12)), implies that $\theta^{k} \le u^{k}(F(x^{i})) + \bar{\delta}^{i}(x^{k} - x^{i})$ ∀x ⁱ∈R ⁿ, because $u^{k}(F(x^{k})) \le u^{k}(F(x^{i})) + \bar{\delta}^{i}(x^{k} - x^{i})$ ∀x ⁱ∈R ⁿ due to the concavity of u ^k(F(x)) (Proposition 4.1). This, together with the assumption that x ^k∈Ω, and the fact that x ^k∈X (since (θ ^k,x ^k) is a solution to R-MOP^k−1), implies that (θ ^k,x ^k) is feasible for the following problem:

$$ \everymath{\displaystyle} \begin{array}{l} \max_{\theta , x}\ \theta\\[6pt] \mbox{s.t.}\quad(1)\ \theta\le u^{k}(F(x^{i})) + \bar{\delta}^{i}(x - x^{i})\quad\forall x^{i} \in R^{n}, \\ \hphantom{\mbox{s.t.}\quad} (2)\ x \in X \cap\varOmega,\quad\theta\in R. \end{array} $$

(A.1)

However, since R-MOP^k is a relaxed version of problem (A.1), then (θ ^k,x ^k) must be optimal for (A.1). Next, problem (A.1) is equivalent to max _x∈X∩Ω u ^k(F(x)), where u ^k(F(x)) (see (12)) is the best approximation of the DM’s preference structure available thus far. Therefore, x ^k is an optimal (“most preferred”) solution for problem MOP. To prove part (b) of the theorem, note that, for any ε>0, x ^t∈X∩Ω is an ε-optimal solution to problem MOP (relative to u ^k(F(x))) if and only if u ^k(F(x ^t))+ε≥u ^k(F(x)) for all x∈X∩Ω. Recalling that LB _k−1=u ^k(F(x ^t)) and UB _k =θ ^k, then LB _k−1+ε≥UB _k implies that u ^k(F(x ^t))+ε≥θ ^k. Now, θ ^k≥u ^k(F(x ^∗)), where (θ ^∗,x ^∗) is a solution to problem (A.1) (see discussion of upper and lower bounds below). Since u ^k(F(x ^∗))≥u ^k(F(x)) ∀x∈X∩Ω, we then obtain that u ^k(F(x ^t))+ε≥u ^k(F(x)) ∀x∈X∩Ω. □

Upper and lower bounds

Let {θ ⁱ,x ⁱ}, i=1,2,…, be the sequence of solutions to successive relaxed master problems R-MOP^k. The sequence {θ ⁱ}, i=1,2,…, is monotone nonincreasing, θ ⁱ⁺¹≤θ ⁱ ∀i, and bounded from below by θ ^∗=u ^k(F(x ^∗)), where (θ ^∗,x ^∗) is an optimal solution to the (unrelaxed) master problem in (A.1). Thus, θ ^∗≤θ ⁱ⁺¹≤θ ⁱ ∀i, and if (θ ^k+1,x ^k+1) is a solution to the most recent problem R-MOP^k, then UB _k+1:=θ ^k+1 is the best upper bound available thus far on the (yet unknown) optimal value u ^k(F(x ^∗)): Step 1 of the algorithm. Consider now the sequence $\{ u_{i}^{k}\}$, i∈K, where $u_{i}^{k} = u^{k}(F(x^{i}))$: it is bounded from above by u ^k(F(x ^∗)), but is not necessarily monotone nondecreasing. Then, $\mathit{LB}_{k}: = u_{t}^{k} = \max\{ u_{i}^{k}: i \in K\}$ is the best lower bound available thus far on u ^k(F(x ^∗)). Consequently, if LB _k +ε≥UB _k+1 for some scalar ε>0, then x ^t is an ε-optimal solution to problem MOP (Theorem 4.2).

Appendix B: Facilitating preference assessments

Local trade-offs

Various procedures may be used to simplify the elicitation of local trade-offs. For example, trade-offs may be obtained by asking the DM to make binary pair-wise comparisons, and a golden-section search may be used. This search is conducted along the line $f_{1}(x^{i}) + \varDelta _{1}^{i}$, and the DM is asked to compare (f ₁(x ⁱ),f _ℓ (x ⁱ)) with $(f_{1}(x^{i}) + \varDelta _{1}^{i}, f_{\ell} (x^{i}) - \varDelta )$ for various values of Δ, until a value $\varDelta _{\ell}^{i}$ is found such that $(f_{1}(x^{i}) , f_{\ell} (x^{i}))\sim(f_{1}(x^{i}) + \varDelta _{1}^{i}, f_{\ell} (x^{i}) - \varDelta _{\ell}^{i})$, where “∼” means “is indifferent to” (Dyer 1973). A golden section search should reduce the number of required pair-wise comparisons that the DM must make. Note that the pair-wise comparisons are binary, in the sense that only two objectives are allowed to change their values while the remaining m−2 objectives are held fixed; this should simplify the task of making trade-offs.

The “willingness-to-pay” technique to rank-order objective vectors

The “willingness-to-pay” technique may be an effective way to rank-order a newly produced objective vector F(x ^k) relative to existing vectors F(x ⁱ), i=1,…,k−1. It has been employed successfully in many multiattribute decision problems: see Keeney and Raiffa (1976) and references therein. Let f _M (x) denote a monetary objective, f ₂(x),…,f _m (x) the remaining objectives, and $(\bar{f}_{2}, \ldots,\bar{f}_{m})$ be a base profile for the m−1 objectives {f ₂,…,f _m }. The base profile $(\bar{f}_{2}, \ldots,\bar{f}_{m})$ is a particular objective vector that the DM feels comfortable comparing/evaluating any other vector $(f_{ 2}^{k}, \ldots,f_{ m}^{k})$ in terms of $(\bar{f}_{2}, \ldots,\bar{f}_{m})$, that is, “pricing-out” $(f_{ 2}^{k}, \ldots,f_{ m}^{k})$ in terms of $(\bar{f}_{2}, \ldots,\bar{f}_{m})$.² The “willingness-to-pay” value Δ ^k is the monetary amount that the DM is just willing to pay to alter $(f_{2}^{k}, \ldots,f_{m}^{k})$ to the base profile $(\bar{f}_{2}, \ldots,\bar{f}_{m})$ (Δ ^k may be positive or negative). Ranking $F^{k} = (f_{M}^{k},f_{2}^{k}, \ldots,f_{m}^{k})$ relative to existing vectors is then done simply by comparing the “willingness-to-pay” value Δ ^k with the already available values Δ ⁱ associated with the existing objective vectors $F^{i} = (f_{M}^{i},f_{2}^{i}, \ldots,f_{m}^{i})$, i=1,…,k−1. Again, DMs may find it relatively easy to think in terms of monetary units and “price out” the attributes {f _2,…, f _m } in terms of the monetary attribute f _M . Also, the assessment of the “willingness-to-pay” value Δ ^k may be simplified by doing it in stages, when only one objective f _ℓ is priced out at a time (Keeney and Raiffa 1976). Finally, computer graphics (including color) can significantly enhance the process of making local trade-offs and pair-wise comparisons.