Bi-objective optimisation over a set of convex sub-problems

Abstract

During the last decades, research in multi-objective optimisation has seen considerable growth. However, this activity has been focused on linear, non-linear, and combinatorial optimisation with multiple objectives. Multi-objective mixed integer (linear or non-linear) programming has received considerably less attention. In this paper we propose an algorithm to compute a finite set of non-dominated points/efficient solutions of a bi-objective mixed binary optimisation problems for which the sub-problems obtained when fixing the binary variables are convex, and there is a finite set of feasible binary variable vectors. Our method uses bound sets and exploits the convexity property of the sub-problems to find a set of efficient solutions for the main problem. Our algorithm creates and iteratively updates bounds for each vector in the set of feasible binary variable vectors, and uses these bounds to guarantee that a set of exact non-dominated points is generated. For instances where the set of feasible binary variable vectors is too large to generate such provably optimal solutions within a reasonable time, our approach can be used as a matheuristic by heuristically selecting a promising subset of binary variable vectors to explore. This investigation is motivated by the problem of beam angle optimisation arising in radiation therapy planning, which we solve heuristically to provide numerical results.

Lagrangian multiplier existence

Assume we have some optimal solution $x^*$ to $\text {MO-FMO}(\varepsilon , \mathscr {A})$ for some $\varepsilon , \mathscr {A}$. We wish to show that optimal dual variables (i.e. Lagrangian multipliers) exist for $\text {MO-FMO}(\varepsilon , \mathscr {A})$ at $x^*$. To show this, we show that solution $x^*$ satisfies the well known linear independence constraint qualification (LICQ), i.e. that the gradients of the active inequality constraints are linearly independent at the optimal solution $x^*$ to $\text {MO-FMO}(\varepsilon , \mathscr {A})$. Note that the constraints $x \in \mathcal {X}(\mathscr {A})$ simply indicate that all fluence variables belonging to beam angles not in $\mathscr {A}$ are zero, and therefore are not present in the problem; see Sect. 5. Therefore (19) effectively has only two gEUD constraints.

If only one of the two constraints is active, then the result follows immediately. Therefore, we focus on the case where both gEUD inequality constraints are satisfied at equality.

Consider some organ $r \in \{T, O_2\}$, and some beamlet k with intensity $x_k$. The kth component in $\nabla {{ gEUD}}^{r}\!(x)$ is given by

$$\begin{aligned} \frac{\partial }{\partial x_k} \textit{gEUD}^{r}\!\left( x\right) = \frac{1}{m^r} \left[ \sum _{j=1}^{m^r} A^r_{jk} \left( \sum _{i=1}^n A^r_{ji} x_i \right) ^{a^r-1} \right] \left[ \frac{1}{m^r} \sum _{j=1}^{m^r}\left( \sum _i^n A^r_{ji} x_i \right) ^{a^{r}}\right] ^{\frac{1}{a^{r}}-1} \end{aligned}$$

(20)

If $\nabla {{ gEUD}}^{T}\!(x^*)$ and $\nabla \text { gEUD}^{O_2}\!(x^*)$ are not linearly independent, then there exists some $h>0$ for which $\frac{\partial }{\partial x_k} \textit{gEUD}^{T}\!\left( x^*\right) = h \cdot \frac{\partial }{\partial x_k} \textit{gEUD}^{O_2}\!\left( x^*\right)$ for all $k=1, 2, \ldots , n$. Our experiments show that each optimal solution $x^*$ to $\text {MO-FMO}(\varepsilon , \mathscr {A})$ has many dozens of non-zero $x^*_i$ values, giving many dozens of non-zero values in $\nabla {{ gEUD}}^{T}\!(x^*)$ and $\nabla \text { gEUD}^{O_2}\!(x^*)$. Furthermore, recall that $A^r_{ji} \geqq 0$ is a real-valued constant describing the rate at which radiation dose along beamlet i is deposited into voxel j in region r, and thus each non-zero $A^r_{ji}$ value is typically unique. Given that we also have $a^T \ne a^{O_2}$, we conclude that for any practical problem, no such h will exist, and so $\nabla {{ gEUD}}^{T}\!(x)$ and $\nabla \text { gEUD}^{O_2}\!(x)$ will be independent, showing that the desired result will hold in practice.