Modified interactive chebyshev algorithm (MICA) for non-convex multiobjective programming

Abstract

In this paper, I carry out an extension of the MICA method (modified interactive chebyshev algorithm) for non-convex multiobjective programming. This method is based on the Tchebychev method and in the reference point approach. At each iteration, the decision maker (DM) can provide aspiration levels (desirable values for the objective functions) and also, if the DM wishes, reservation levels (level under which the objective function is not considered acceptable). On the basis of this preferential information, a region of the nondominated objective set is defined. In the convex case, considering the aspiration vector as a reference point in an achievement scalarizing function and taking a set of weight vectors, the efficient solutions generated satisfy the reservation levels. In this work, I analyze the non-convex case. The main result of MICA is verified and demonstrated for the non-convex bi-objective case. The MICA method is not verified in general for multiobjective problems with three or more objective functions, which is demonstrated with a counterexample.

Appendix

In this appendix, I prove Theorem 2 (stated in Sect. 4):

Proof of Theorem 2

Given that $(\bar{\mathbf{x}}, \bar{\mathbf{z}})$ is an optimal solution of problem (4), the following relation holds:

$$\begin{aligned} s(\mathbf{q}^{h}, \bar{\mathbf{z}}, \bar{\varvec{\lambda }}) \le s(\mathbf{q}^{h}, \mathbf{z}^{hr}, \bar{\varvec{\lambda }}) \quad (r=1,2). \end{aligned}$$

(11)

For each $i \in \{1,2\}$ and $r \in \{1,2\}$, taking into account the definitions of $\varvec{\lambda }^{hr}$ in (9), and $\delta$, we have:

$$\begin{aligned} \text {If } i \in J^{hr} \quad \Rightarrow \quad q_{i}^{h} - z_{i}^{hr} = \frac{C^{hr}}{\lambda _{i}^{hr}}, \\ \text {If } i \in I^{hr} \quad \Rightarrow \quad q_{i}^{h} - z_{i}^{hr} \le 0 < \delta = \frac{C^{hr}}{\lambda _{i}^{hr}}. \end{aligned}$$

Therefore, inequality (11) implies that

$$\begin{aligned} \max _{i=1,2} \left\{ \bar{\lambda }_{i} (q_{i}^{h} - \bar{z}_{i}) \right\} \le \max _{i=1,2} \left\{ \bar{\lambda }_{i} (q_{i}^{h} - z_{i}^{hr}) \right\} \le \max _{i=1,2} \left\{ \bar{\lambda }_{i} \frac{C^{hr}}{\lambda _{i}^{hr}} \right\} \le C^{hr} \max _{i=1,2} \left\{ \frac{\bar{\lambda }_{i}}{\lambda _{i}^{hr}} \right\} \end{aligned}$$

and thus

$$\begin{aligned} \max _{i=1,2} \left\{ \bar{\lambda }_{i} (q_{i}^{h} - \bar{z}_{i}) \right\} \le C^{hr} \max _{i=1,2} \left\{ \frac{\bar{\lambda }_{i}}{\lambda _{i}^{hr}} \right\} \end{aligned}$$

(12)

Without loss of generality, it can be assumed that $\lambda _{1}^{h1} \le \lambda _{1}^{h2}$, which in turn implies that $\lambda _{2}^{h1} \ge \lambda _{2}^{h2}$ (given that $\lambda _{1}^{h1} + \lambda _{2}^{h1} = \lambda _{1}^{h2} + \lambda _{2}^{h2} =1$). Therefore, taking into account expression (9), we have:

$$\begin{aligned} \lambda _{1}^{h1} \le \bar{\lambda }_{1} \le \lambda _{1}^{h2} \qquad \text {and} \qquad \lambda _{2}^{h2} \le \bar{\lambda }_{2} \le \lambda _{2}^{h1} \Rightarrow \\ \Rightarrow \frac{\bar{\lambda }_{1}}{\lambda _{1}^{h1}} \ge 1 \ge \frac{\bar{\lambda }_{2}}{\lambda _{2}^{h1}} \qquad \text {and} \qquad \frac{\bar{\lambda }_{2}}{\lambda _{2}^{h2}} \ge 1 \ge \frac{\bar{\lambda }_{1}}{\lambda _{1}^{h2}} \Rightarrow \\ \Rightarrow \max _{i=1,2} \left\{ \frac{\bar{\lambda }_{i}}{\lambda _{i}^{hr}} \right\} = \frac{\bar{\lambda }_{r}}{\lambda _{r}^{hr}} \quad (r=1,2). \end{aligned}$$

Therefore, relation (12) implies:

$$\begin{aligned}&\max _{i=1,2} \left\{ \bar{\lambda }_{i} (q_{i}^{h} - \bar{z}_{i}) \right\} \le C^{hr} \max _{i=1,2} \left\{ \frac{\bar{\lambda }_{i}}{\lambda _{i}^{hr}} \right\} = C^{hr} \frac{\bar{\lambda }_{r}}{\lambda _{r}^{hr}} \quad (r=1,2) \Rightarrow \nonumber \\&\Rightarrow \bar{\lambda }_{r} (q_{r}^{h} - \bar{z}_{r}) \le C^{hr} \frac{\bar{\lambda }_{r}}{\lambda _{r}^{hr}} \quad (r=1,2) \Rightarrow \nonumber \\&\Rightarrow q_{r}^{h} - \bar{z}_{r} \le \frac{C^{hr}}{\lambda _{r}^{hr}} \quad (r=1,2) . \end{aligned}$$

(13)

If, on the one hand, $r \in J^{hr}$, then relations (9) and (13) imply that:

$$\begin{aligned}&q_{r}^{h} - \bar{z}_{r} \le \frac{C^{hr}}{\lambda _{r}^{hr}} = q_{r}^{h} - z_{r}^{hr} \quad (r=1,2) \Rightarrow \\&\Rightarrow \bar{z}_{r} \ge z_{r}^{hr} \ge \varepsilon _{^r}^{h} \quad (r=1,2). \end{aligned}$$

If, on the other hand, $r \in I^{hr}$, then relations (9) and (13) imply that:

$$\begin{aligned}&q_{r}^{h} - \bar{z}_{r} \le \frac{C^{hr}}{\lambda _{r}^{hr}} = \delta \le q_{r}^{h} - \varepsilon _{^r}^{h} \quad (r=1,2) \Rightarrow \\&\Rightarrow \bar{z}_{r} \ge \varepsilon _{^r}^{h} \quad (r=1,2), \end{aligned}$$

and this completes the proof. $\square$