A vector linear programming approach for certain global optimization problems

TL;DR

The modification of the MOLP algorithms results in a more efficient treatment of the studied optimization problems, and various other classes of global optimization problems can be expressed in this way.

Abstract

Global optimization problems with a quasi-concave objective function and linear constraints are studied. We point out that various other classes of global optimization problems can be expressed in this way. We present two algorithms, which can be seen as slight modifications of Benson-type algorithms for multiple objective linear programs (MOLP). The modification of the MOLP algorithms results in a more efficient treatment of the studied optimization problems. This paper generalizes results of Schulz and Mittal on quasi-concave problems and Shao and Ehrgott on multiplicative linear programs. Furthermore, it improves results of Löhne and Wagner on minimizing the difference $f=g-h$ of two convex functions $g$, $h$ where either $g$ or $h$ is polyhedral. Numerical examples are given and the results are compared with the global optimization software BARON.

Figures (4)

• Figure 1: $P[S]$ and level sets of $f$ for Example \ref{['example_pa']}. The cone $C$ generated by $(-1,0)^\intercal$ and $(0,1)^\intercal$ is indicated by the dashed lines. It apparently reflects the objective's monotonicity within the feasible region.
• Figure 2: Iteration steps for Example \ref{['example_pa']}. The white circle shows the current point $t$, whereas the gray dots indicate the boundary points $s$ calculated by \ref{['scl:p2']}. Note that we avoid two additional iteration steps in comparison to Algorithm \ref{['alg_primal_benson']} since there are two remaining vertices of the outer approximation, which do not need to be processed.
• Figure 3: Visualization of the initialization and the first two iteration steps of Algorithm \ref{['alg_dual_modification']} by Example \ref{['example_pa_dual']}. The shrinking sequence $\mathcal{O}^*_j$ of outer approximations of $\mathcal{D}^*$ corresponds to the expanding sequence $\mathcal{I}_j \mathrel{\vcenter{\hbox{\scriptsize.}\hbox{\scriptsize.}}}= \{ y \in \mathbb{R}^q \mid \forall y^* \in \mathcal{O}^*_j:\; \varphi(y,y^*)\geq 0\}$ of inner approximations of $\mathcal{P}$ by geometric duality. Likewise, there is an expanding sequence $\mathcal{I}_j^* \mathrel{\vcenter{\hbox{\scriptsize.}\hbox{\scriptsize.}}}= \{ y^* \in \mathbb{R}^q \mid \forall y \in \mathcal{O}_j:\; \varphi(y,y^*)\geq 0\}$ of inner approximations of $\mathcal{D}^*$ corresponding to the shrinking sequence of outer approximations $\mathcal{O}_j$ of $\mathcal{P}$. The white circle indicates the vertex of $\mathcal{O}_j$ chosen as $t$ in line \ref{['alg4_argmin']}. The black dots label the corresponding points $t^*$, see line \ref{['alg4_vsr']}. Again, the gray dots indicate the calculated boundary points. The calculations are based on the choice of $c=(-0.25 , 1)^\intercal$ as inner point of $C$. Notice that even though we have $\mathcal{D}^*=\mathcal{O}^*_2$, the algorithm does not terminate after two iterations because $t$ is not an element of $\mathcal{P}$. Another two iteration steps are required to identify the problems solution $(1.084,0.804)^\intercal$.
• Figure 4: Image $P[S]$ and upper image $\mathcal{P}$ of \ref{['vlp']} and upper image $\mathcal{M}$ of \ref{['vlp2']} for Example \ref{['ex_nonsolid']}. Notice the facet in $\mathcal{M}$ corresponding to $\mathcal{P}$.

Theorems & Definitions (38)

• Example 1: DC programming - "convex component" being polyhedral
• Example 2: DC programming - "concave component" being polyhedral
• Example 3: Minimizing a convex function over the boundary of a polytope
• Example 4: Linear multiplicative programming linear_multiplicative
• Proposition 5
• proof
• Corollary 6
• proof
• Proposition 7: benson_type
• Theorem 8: see benson_type