A deterministic optimization algorithm for nonconvex and combinatorial bi-objective programming

Abstract

any practical multiobjective optimization (MOO) problems include discrete decision variables and/or nonlinear model equations and exhibit disconnected or smooth but nonconvex Pareto surfaces. Scalarization methods, such as the weighted-sum and sandwich (SD) algorithms, are common approaches to solving MOO problems but may fail on nonconvex or discontinuous Pareto fronts. In the current work, motivated by the well-known normal boundary intersection (NBI) method and the SD algorithm, we present SDNBI, a new algorithm for bi-objective optimization (BOO) designed to address the theoretical and numerical challenges associated with the reliable solution of general nonconvex and discrete BOO problems. The main improvements in the algorithm are the effective exploration of the nonconvex regions of the Pareto front and, uniquely, the early identification of regions where no additional Pareto solutions exist. The performance of the SDNBI algorithm is assessed based on the accuracy of the approximation of the Pareto front constructed over the disconnected nonconvex objective domains. The new algorithm is compared with two MOO approaches, the modified NBI method and the SD algorithm, using published benchmark problems. The results indicate that the SDNBI algorithm outperforms the modified NBI and SD algorithms in solving convex, nonconvex-continuous, and combinatorial problems, both in terms of computational cost and of the overall quality of the Pareto-optimal set, suggesting that the SDNBI algorithm is a promising alternative for solving BOO problems.

Figures (17)

• Figure 1: Illustration of a nonconvex Pareto front and the generation of an inner approximation that cuts off part of the front. The grey curve represents the boundary of the feasible region in bi-objective space for problem MOP1. The black dashed line is a facet obtained by generating a convex hull using $\boldsymbol{z}^{A1}$ and $\boldsymbol{z}^{A2}$. The nondominated point is $\boldsymbol{z}^1$ generated by the solution of \ref{['eq:ch6:mNBI']} with the normal vector $\boldsymbol{\bar{n}}^1$ and the reference point $\boldsymbol{\Phi}^{1}\boldsymbol{\beta}^{1}$. The corresponding line $H_2(\boldsymbol{\bar{n}}^1, b^1)$ (blue dashed line) is not tangent to the Pareto front. If the next upper bounds, i.e., inner approximations, are obtained by constructing the convex hull based on the updated set of the Pareto points, $\boldsymbol{Z_E}=\{\boldsymbol{z}^{A2}, \boldsymbol{z}^{1}, \boldsymbol{z}^{A1}\}$, the solid black lines are obtained, cutting off the part of the Pareto front shown in orange.
• Figure 2: Representation of the line $H_2(\boldsymbol{w}'^1,b^1)$ passing through $\boldsymbol{z}^1$ with normal vector $\boldsymbol{w}'^1$ (red solid line), calculated based on Equation \ref{['eq:ch6:normal-mNBI']}. The grey curve represents the boundary of the feasible region in bi-objective space and the black solid lines are the facets obtained by constructing a convex hull based on the current Pareto points $\boldsymbol{z}^{A1}$, $\boldsymbol{z}^{A2}$ and $\boldsymbol{z}^{1}$. Facet $(\boldsymbol{z}^{1},\boldsymbol{z}^{A1})$ and line $H_2(\boldsymbol{w}'^1,b^1)$ cannot be used to construct approximations of the Pareto front due to nonconvexity.
• Figure 3: Schematic illustrating the procedure for the decomposition of the bi-objective space in the case of a nonconvex Pareto front, with boundary points of the feasible region shown with grey symbols ($\bullet$): (a) An initial objective space is characterized by subspace $C^0(\boldsymbol{z}^{A2},\boldsymbol{z}^{A1}$) (black rectangular box), assumed to contain a convex Pareto front. A new nondominated point $\boldsymbol{z}^1$ is found by solving problem \ref{['eq:ch6:mNBI']} for $\boldsymbol{\Phi}^{1}\boldsymbol{\beta}^{1}$; (b) the initial objective space is decomposed into subspaces $C^1(\boldsymbol{z}^{1},\boldsymbol{z}^{A1})$ and $C^2(\boldsymbol{z}^{A2},\boldsymbol{z}^{1})$ that are assumed to contain nonconvex and convex Pareto fronts, respectively. The inner approximation of the Pareto front in $C^2$ is the facet $(\boldsymbol{z}^{A2},\boldsymbol{z}^{1})$ (black dashed line) and the outer approximation is given by the segment of the tangent at $\boldsymbol{z}^{1}$ (red solid line) connecting $\boldsymbol{z}^{1}$ to the $f_1=0$ line and by the $f_1=0$ line. For subspace $C^1$, the inner approximation is given by the segment of the tangent at $\boldsymbol{z}^{1}$ (red solid line) connecting $\boldsymbol{z}^{1}$ to the $f_1=1$ line and by the $f_1=1$ line, while the outer approximation is given by the facet $(\boldsymbol{z}^{1},\boldsymbol{z}^{A1})$; (c) the approximation of the Pareto front is improved by adding $\boldsymbol{z}^2$ and the subspace $C^1$ is not decomposed since $\boldsymbol{w}'^{2 \top}\boldsymbol{z} \leq b^{2}$ holds for all $\boldsymbol{z}\in \boldsymbol{Z}^{C^1}_{\boldsymbol{E}}$. For subspace $C^1$, the inner approximation is given by the segment of the tangent at $\boldsymbol{z}^{2}$ (red solid line), while the outer approximation is given by the convexhull $(\boldsymbol{z}^{1},\boldsymbol{z}^{2},\boldsymbol{z}^{A1})$
• Figure 4: A geometrical illustration of the solution of the \ref{['eq:ch6:mNBI']} subproblem for a disconnected Pareto front. The boundary points of the feasible region are shown with grey symbols ($\bullet$). We begin by considering subspace $C^3(\boldsymbol{z}^4,\boldsymbol{z}^5)$ (black rectangular box). The solution of the \ref{['eq:ch6:mNBI']} subproblem for facet $F_{S}(\boldsymbol{z}^4,\boldsymbol{z}^5)$ and $\boldsymbol{\beta}^6$ lies at $\boldsymbol{z}^4$. The blue shaded region represents the area where $\boldsymbol{\Phi}^6\boldsymbol{\beta}^6 + t^*\boldsymbol{\bar{n}}^6 \ge \boldsymbol{z}^4-\boldsymbol{f}^{id}$.
• Figure 5: A geometrical illustration of the SDNBI procedure for a disconnected Pareto front when using an alternative subproblem \ref{['eq:ch6:simpleNBIn']}. The boundary points of the feasible region are shown with grey symbols ($\bullet$). (a) A new Pareto point $\boldsymbol{z}^6$ is generated as a solution of \ref{['eq:ch6:simpleNBIn']}. The red shaded region represents the area where ($\boldsymbol{\Phi}^6\boldsymbol{\beta}^6 + t'^*\boldsymbol{\bar{n}}^6 \ge \boldsymbol{z}^6-\boldsymbol{f}^{id}) \cap (z^6_1 \geq z^4_1 + \epsilon_z$). The solid red line at $\boldsymbol{z}^6$ represents a tangent at the solution. The facet $F_{S}(\boldsymbol{z}^4,\boldsymbol{z}^6)$ generated by an inner approximation is discarded from the search space in the subsequent iterations. In (b), it can be seen that any choice of reference points on the line segment defined by $\theta\boldsymbol{z}^4 +(1-\theta) \boldsymbol{\Phi}^6\boldsymbol{\beta}^{'6}$, $\forall \ 0 < \theta \leq1$ (shown by the blue shaded region), produces the same solution $\boldsymbol{z}^4$.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Definition 9
• Remark 1