Diagonal Generalizaton of the DIRECT Method for Problems with Constraints

Abstract

The DIRECT method solves Lipschitz global optimization problems on a hyperinterval with an unlimited range of Lipschitz constants. We propose an extension of the DIRECT method principles to problems with multiextremal constraints is proposed when two evaluations of functions at the ends of the chosen main diagonals are used at once. We present computational illustrations, including the solution of a problem with discontinuities. We also perform convergence analysis.

Appendix

Proof of Statement. If conditions (24) are satisfied for some ΔL from the specified interval, then (21), (22), obviously, will hold with the same ΔL and with ΔL^g = α ΔL. Now let (21), (22) be satisfied and suppose that for D_t there exists a corresponding pair $\widetilde{\Delta L}$, ${\widetilde{\Delta L}}^{g}$. But since g⁻(ΔL^g, D_i) is monotone decreasing with increasing ΔL^g, then (21), (22) will certainly hold for the values $\widetilde{\Delta L}$, $\Delta {L}^{g}=\alpha \widetilde{\Delta L}$, as needed. This completes the proof of the statement.

Proof of Lemma. At the first stage of processing the search information, at iteration k, as described in Section 3, in each d-layer at least one modified non-dominated hyperinterval ${\widehat{D}}_{s}^{d}$ of this layer is selected. The last selected hyperinterval has an associated range $[\Delta {L}_{s}^{d},\Delta {L}_{s+1}^{d})$ with the largest number s = S_k(d), where $\Delta {L}_{{S}_{k}(d)+1}^{d}=+\infty$. This holds, in particular, for the d-layer of hyperintervals with the largest diameter $d={d}_{k}^{\max }$ at the current kth iteration.

Further, when constructing the partition (25) of the axis of increments ΔL in the last interval [ΔL_j, ΔL_j+1) with number j = m_k, the value $\Delta {L}_{{m}_{k}+1}=+\infty$. This interval corresponds to the set of hyperintervals ${\widetilde{{\mathcal{O}}}}_{k}^{{m}_{k}}$, which by construction necessarily includes at least one hyperinterval ${\widehat{D}}_{s}^{d}$ on the d-layer with $d={d}_{k}^{\max }$ (i.e., maximum diameter) corresponding to the range $[\Delta {L}_{s}^{d},\Delta {L}_{s+1}^{d})$ of values ΔL with the number $s={S}_{k}({d}_{k}^{\max })$. On the comparison plane (d/2, F), when comparing hyperintervals from ${\widetilde{{\mathcal{O}}}}_{k}^{{m}_{k}}$, this hyperinterval will correspond to the rightmost point in the resulting set of points ${P}_{k}^{{m}_{k}}$. Since the comparison is done for the interval $[\Delta {L}_{{m}_{k}},+\infty )$, the point with the largest $d={d}_{k}^{\max }$ with a sufficiently large ΔL will surely dominate the others and condition (17) will also be fulfilled at this point. Therefore, the hyperinterval of the largest diameter will be included in the set ${\widehat{{\mathcal{D}}}}_{k}$ of hyperintervals partitioned on iteration k. This completes the proof of the lemma.

Proof of Theorem. Due to the assumption of global stability of the set of solutions X^* (condition C), as well as condition B to the structure of the set of global minima X^* and the functions of the problem, to prove the theorem it suffices to prove that the test point placement is everywhere dense in the limit. Indeed, the search set D is compact. If conditions A and B are met, the test point placement which is everywhere dense in the limit leads to the appearance of subsequences of test points that converge to each of the solutions x^*. Each x^* belongs to the closure of some open subset χ from X, if $X\ne \varnothing$, or from D otherwise. Given a test point placement which is everywhere dense in the limit, there are points arbitrarily close to x^* that are centers of open balls included in χ. In each of these balls, the method will place a test point at some point. Therefore, there are subsequences of admissible points from subsets χ that converge to x^* with an unlimited increase in the number of iterations. Functions Q, g or (with $X=\varnothing$) the function g are continuous on the closures $\overline{\chi }$. Thus, the sequence of the points with the best function evaluations ${x}_{k}^{* }$ will be minimizing and, by virtue of requirement C, its limit points will represent solutions to the problem.

It remains to prove that the functions evaluation points distribution is everywhere dense in the limit. We give only a sketch of the proof. The desired behavior follows from the lemma, which establishes that in the constructed method at least one of the intervals with largest diameter is necessarily partitioned at each iteration. Since the partitioning of hyperintervals occurs in three equal parts along the largest edge, this ensures a strict decrease in diameter with a certain coefficient separated from unity. These two factors are sufficient. A more detailed argument can be constructed similarly to the proof of Theorem 5.6 from [8], taking into account the lemma. This completes the proof of the theorem.