Optimizing a 2D Function Satisfying Unimodality Properties

A general formulation of discrete optimization is to maximize a given function f : D → R over a discrete (finite) domain D. In general, of course, this problem may require |D| probes to f . One approach to making optimization more tractable is to be satisfied with finding a local maximum, i.e., a point at which f attains a value larger than all “neighboring” points, for some definition of neighborhoods. In particular, for the standard 1D domain D = {1, 2, . . . , n}, Fibonacci search [Kie53] finds a local maximum using logφ n + O(1) probes, where φ = (1+ √ 5)/2 is the golden ratio. Surprisingly, even for a square 2D domain D = {1, 2, . . . , n} × {1, 2, . . . , n}, the problem complexity grows exponentially: Mityagin [Mit03] proved that Θ(n) probes to such an f are sufficient and sometimes necessary. Thus weakening the problem to finding local maxima does not cause the exponential speedup from 1D in higher dimensions. Another approach to making optimization more tractable is to add assumptions about the function f . For example, if we assume that f is unimodal (denoted “ ̄ unimodal”), i.e., it has exactly one local maximum, then finding local maxima and finding global maxima are equivalent. One could hope that having this structural information about the function would also help in finding that maximum. Unfortunately, a careful reading of the construction in [Mit03] of 2D functions f requiring Θ(n) probes are in fact unimodal. We study the related condition that the 2D function f is unimodal in every row (↔ unimodal) and/or in every column (l unimodal). (These properties are satisfied by e.g. convex functions.) While seemingly weaker than unimodality, these properties are incomparable to unimodality, and in fact result in exponential speedup for finding local maxima. Table 1 summarizes all of our results. Our upper bounds all follow from a combination of linear search and/or Fibonacci search in each dimension. Matching local bounds for global optimization follow in some cases from independence of the columns. Some bounds are tight only up to logarithmic factors, leaving intriguing open questions. In the full paper, we provide the omitted proof and prove the comforting fact that a natural random probing algorithm makes Ω(lgm lg n) expected probes even for a convex function, as in our dual Fibonacci search. Other properties of matrices facilitating optimization have been studied; see [GP92].