Efficient micro immune optimization approach solving constrained nonlinear interval number programming

Abstract

This work investigates a possibility degree-based micro immune optimization approach to seek the optimal solution of nonlinear interval number programming with constraints. Such approach is designed under the guideline of the theoretical results acquired in the current work, relying upon interval arithmetic rules, interval order relation and immune theory. It involves in two phases of optimization. The first phase, based on a new possibility degree approach, assumes searching efficient solutions of natural interval extension optimization. This executes five modules - constraint bound handling, population division, dynamic proliferation, mutation and selection, with the help of a varying threshold of interval bound. The second phase collects the optimal solution(s) from these efficient solutions after optimizing the bounds of their objective intervals, in terms of the theoretical results. The numerical experiments illustrated that such approach with high efficiency performs well over one recent nested genetic algorithm and is of potential use for complex interval number programming.

Appendix A

Proof of Lemma 1.

For a given x ∈ D, let Θ _i (x) represent the set of optimal solutions of the constraint function g _i over U ^I, i.e., ${\Theta }_{i}(x) = \{u^{\ast }\in U^{I}| g_{i}(x,u^{\ast }) = \underset {u}{min} g_{i}(x,u)\}.$ Since g _i (x, u) is continuous in u over U ^I, Θ _i (x) is nonempty. Define a set-valued mapping $g_{i}(.,{\Theta }_{i}(.)): D\longrightarrow 2^{R^{p}}$, satisfying g _i (x,Θ _i (x)) = {g _i (x, u ^∗)|u ^∗ ∈ Θ _i (x)}. Since g _i is continuous over D × U ^I, Θ _i (.) is upper semi-continuous in x according to the set-valued analysis theory in mathematics; accordingly, g _i (.,Θ _i (.)) is also upper semi-continuous in x. Again since g _i (x,Θ _i (x)) is a singleton set, it is continuous in x ∈ D. Hence, $\underline {g}_{i}$ is continuous in x, with $g_{i}(x,{\Theta }_{i}(x)) = \{\underline {g}_{i}(x)\}$. Similarly, one can prove that $\overline {g}_{i}(.)$, $\underline {h}_{j}(.)$ and $\overline {h}_{j}(.)$ are also continuous. Consequently, the conclusion is true by means of (13). □

Proof of Lemma 2.

Take x ^∗ ∈ Γ _{s o} . By definition 1, if x ^∗ ∉ Γ, there exists y ∈ Γ such that F(y) < F(x ^∗), i.e., $\overline {f}(y)<\underline {f}(x^{\ast })$. Again since $\overline {f}(x^{\ast }) = \overline {f}^{\ast }$ and $\underline {f}(x^{\ast }) = \underline {f}^{\ast }$, we derive $\overline {f}^{\ast }=\overline {f}(x^{\ast })\leq \overline {f}(y)<\underline {f}^{\ast }$. This results in contradiction, and thus we have Γ _{s o} ⊆ Γ. On the other hand, Take x ^∗ ∈ Γ. If x ^∗ ∉ Λ, there exists y ∈ Λ such that π(y) < π(x ^∗), i.e., f ^R(y) < f ^L(x ^∗). Again since F(x) ⊆ π(x), we have $\overline {f}(y)\leq f^{R}(y)<f^{L}(x^{\ast })\leq \underline {f}(x^{\ast }),$ which derives F(y) < F(x ^∗). Therefore, the contradiction is yielded, and hence we gain Γ ⊆ Λ. □

Proof of Lemma 3.

Owing to Γ ⊆ Σ, we gain $\underline {f}^{\ast }\leq \underline {F}^{\ast }$ and $\overline {f}^{\ast }\leq \overline {F}^{\ast }$. Take x ^∗ ∈ Σ satisfying $\overline {f}(x^{\ast }) = \overline {f}^{\ast }$. If x ^∗ ∉ Γ, by means of definition 1 there exists y ∈ Γ such that F(y) < F(x ^∗), i.e., $\overline {f}(y)<\underline {f}(x^{\ast })$, which follows $\underline {f}(y)<\underline {f}(x^{\ast })$. This yields contradiction, due to y ∈ Σ. Thereby, we gain x ^∗ ∈ Γ, and hence have $\underline {f}^{\ast }\geq \underline {F}^{\ast }$. This way, it is true that $\underline {F}^{\ast }=\underline {f}^{\ast }$. Similarly, we can know that the equality of $\overline {F}^{\ast }=\overline {f}^{\ast }$ is true. □

Proof of Theorem 1.

• (a) by (14), there exists x ^∗ ∈ Σ such that f ^R(x ^∗) = σ. If x ∈ Λ but f ^L(x) > σ, then we get f ^R(x ^∗) < f ^L(x), i.e., π(x ^∗) < π(x). This is not consistent with x ∈ Λ. Thereby, we obtain f ^L(x) ≤ σ. On the other hand, if f ^L(x) ≤ σ but x ∉ Λ, there exists y ∈ Λ satisfying π(y) < π(x), i.e., f ^R(y) < f ^L(x). Accordingly, we have f ^R(y) < σ. This yields contradiction by means of (14). Consequently, the conclusion (a) holds.

• (b) Let x ∈ Γ; if x ∉ Λ₀, then there exists y ∈ Λ satisfying F(y) < F(x), due to Γ ⊆ Λ. This results in contradiction, owing to x, y ∈ Σ. Hence, we get Γ ⊆ Λ₀. Conversely, take x ∈ Λ₀. If x ∉ Γ, then there exists y ∈ Γ satisfying F(y) < F(x). Since Γ ⊆ Λ and Λ₀ ⊆ Λ, we have x, y ∈ Λ. Accordingly, the contradiction is yielded because of F(y) < F(x). All these illustrate that the conclusion (b) is true.

• (c) let x ^∗ ∈ F _{s o} . By Lemma 2, we acquire that x ^∗ is also an efficient solution for P _{I V} , i.e., x ^∗ ∈ Γ. Again, by Lemma 3, we get x ^∗ ∈ Γ _o , and accordingly Γ _{s o} ⊆ Γ _o . Conversely, by Lemma 3, it is obvious that Γ _o ⊆ Γ _{s o} . Therefore, the conclusion (c) is true.

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Proof of Lemma 4.

When calculating the upper or lower bound of a given uncertain constraint (g _i or h _j , 1 ≤ i ≤ J, 1 ≤ j ≤ K) with given x ∈ A, step 5 creates l subclasses with complexity O(M l o g M); step 9 mutates all the clones with complexity O(M q); step 10 chooses M cells to constitute the new population with computational complexity O(M ²). Accordingly, the complexity is O(N(M q + M ²)), and thus the conclusion is true. □

Proof of Theorem 2.

Through the algorithm formulation, μCIOA’s complexity is decided by steps 8, 9, 12 and 16. In step 8, step 8.1 or 8.2 executes mutation with N p runs; Lemma 4 follows that step 8.4 decides the bounds of uncertain constraints with complexity O(N(J + K)(M q + M ²)). Thus, the complexity of step 8 is decided by O(N(J + K)(M q + M ²) + N p). In step 9, step 9.1 runs with the complexity of O(N l o g N); step 9.2 performs mutation with N p times; the complexity of step 9.3 is O(N(J + K)(M q + M ²)). Thus, step 9 is with the complexity of O(N(J + K)(M q + M ²) + N l o g N + N p). In step 12, the complexity, decided by the crowding distance method is O(N ²). By means of Lemma 4, step 16 is with complexity O(N(M q + M ²)) in the worst case. Summarily, μCIOA’s complexity O _c in the worst case is determined by

$$\begin{array}{@{}rcl@{}} O_{c}&=&(N(J+K)(Mq+M^{2})+Np)\\ &&+O(N(J+K)(Mq+M^{2})+NlogN+Np)\\ &&+O(N^{2})+O(N(Mq+M^{2}))\\ &=&O(N(J+K)(Mq+M^{2})+N^{2}+Np). \end{array} $$

(18)

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