Transformation of Quasiconvex Functions to Eliminate Local Minima

Suliman Al-Homidan (), Nicolas Hadjisavvas () and Loai Shaalan ()
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Suliman Al-Homidan: King Fahd University of Petroleum and Minerals
Nicolas Hadjisavvas: King Fahd University of Petroleum and Minerals
Loai Shaalan: King Fahd University of Petroleum and Minerals

Journal of Optimization Theory and Applications, 2018, vol. 177, issue 1, No 4, 93-105

Abstract: Abstract Quasiconvex functions present some difficulties in global optimization, because their graph contains “flat parts”; thus, a local minimum is not necessarily the global minimum. In this paper, we show that any lower semicontinuous quasiconvex function may be written as a composition of two functions, one of which is nondecreasing, and the other is quasiconvex with the property that every local minimum is global minimum. Thus, finding the global minimum of any lower semicontinuous quasiconvex function is equivalent to finding the minimum of a quasiconvex function, which has no local minima other than its global minimum. The construction of the decomposition is based on the notion of “adjusted sublevel set.” In particular, we study the structure of the class of sublevel sets, and the continuity properties of the sublevel set operator and its corresponding normal operator.

Keywords: Quasiconvex function; Generalized convexity; Adjusted sublevel sets; Normal operator; 90C26; 90C30; 26A51; 26B25 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (6)

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DOI: 10.1007/s10957-018-1223-7

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