Abstract
We suggest an approach to solve special classes of multi-extreme problems to optimize the combination (e.g., sum, product) of several functions, under the assumption that the effective algorithms to optimize each of this item are known. The algorithm proposed is iterative. It realizes one of the idea of the branch-and-bound method and consists in successive correcting of the low and the upper bounds of optimal value of objective functions. In each iteration, the total area of the considered region that may contain the image optimal point, decreases at least twice. Various techniques that accelerate the process of finding solutions are discussed.
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Notes
One of the most popular classes of such problems is, for example, the minimization of quasi-convex functions. Of course, quasi-convex function is in general a multiextremal (with flat local minima) and even discontinuous function. Hence, global minimization of a quasi-convex function can be a hard optimization problem. In [23] Konnov considered a normalized subgradient method for minimization of quasi-convex functions. This algorithm is a modification of the algorithm by Polyak [25] for convex functions. Also, let us mention the works [12, 17]. Another, more restricted classes of generalized convex functions which admit the effective optimization algorithms, are semi-strictly quasi-convex, strictly quasi-convex, pseudo-convex, strictly pseudo-convex, strongly pseudo-convex. The exact definitions can be found e.g. in [2] (see also [15]) where the relationships between families of convex and generalized convex functions are illustrated by the beautiful figure (see p. 46) and examples (see p. 47).
Note that here we consider convergence “in functional”, i.e., search mainly the optimal value of the functional, not the optimal point.
Note that \( \underline{\nu _1} \le \nu _1^* \le \overline{\nu _1}, \ \ \underline{\nu _2} \le \nu _2^* \le \overline{\nu _2}, \ \ \underline{\nu } \le \nu _1^*+\nu _2^* \le \overline{\nu }. \)
This means that the case shown in Fig. 3 is realized.
This means that the case shown in Fig. 6 is realized.
If the situation shown in Fig. 8, are realized, then delete the triangles (or parts thereof) located “between” the two hypotenuses. Renumber the remaining triangles. Recalculate for each of the triangles the value y (since in the triangles the shaded part was removed).
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The author would like to thank the anonymous referees for very valuable comments.
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The work was supported by the program of fundamental scientific researches of the SB RAS No. I.5.1., Project No. 0314-2016-0018. Supported in part by RFBR Grants 16-01-00108, 16-06-00101 and 18-010- 00728.
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Bykadorov, I. Acceleration procedure for special classes of multi-extremal problems. Optim Lett 13, 1819–1835 (2019). https://doi.org/10.1007/s11590-018-1355-6
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DOI: https://doi.org/10.1007/s11590-018-1355-6