Minimizing a sum of clipped convex functions
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Minimizing a sum of clipped convex functions Shane Barratt1 · Guillermo Angeris1 · Stephen Boyd1 Received: 28 October 2019 / Accepted: 28 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider the problem of minimizing a sum of clipped convex functions. Applications of this problem include clipped empirical risk minimization and clipped control. While the problem of minimizing the sum of clipped convex functions is NP-hard, we present some heuristics for approximately solving instances of these problems. These heuristics can be used to find good, if not global, solutions, and appear to work well in practice. We also describe an alternative formulation, based on the perspective transformation, that makes the problem amenable to mixed-integer convex programming and yields computationally tractable lower bounds. We illustrate our heuristic methods by applying them to various examples and use the perspective transformation to certify that the solutions are relatively close to the global optimum. This paper is accompanied by an open-source implementation. Keywords Convex optimization · Mixed-integer programming · Robust statistics
1 Introduction Suppose f : Rn → R is a convex function, and α ∈ R. We refer to the function min{ f (x), α} as a clipped convex function. In this paper we consider the problem of minimizing a sum of clipped convex functions,
minimize f 0 (x) +
m
min{ f i (x), αi },
(1)
i=1
B
Shane Barratt [email protected] Guillermo Angeris [email protected] Stephen Boyd [email protected]
1
Department of Electrical Engineering, Stanford University, Stanford, USA
123
S. Barratt et al.
with variable x ∈ Rn , where f 0 : Rn → R∪{+∞} and f i : Rn → R for i = 1, . . . , m are closed proper convex functions [19], and αi ∈ R for i = 1, . . . , m. We use infinite values of f 0 to encode constraints on x, i.e., to constrain x ∈ X for a closed convex / X . When f i (x) > αi , the value of the ith term set X , we let f 0 (x) = +∞ for all x ∈ in the sum is clipped to αi , which limits how large each term in the objective can be. Many practical problems can be formulated as instances of (1); we describe a few in Sect. 2. NP-hardness. In general, problem (1) is nonconvex and as a result can be very difficult to solve. Indeed, it is NP-hard. We show this by giving a reduction of the subset sum problem to an instance of (1). The subset sum problem involves determining whether or not there exists a subset of a given set of integers a1 , . . . , an that sum to zero. The optimal value of the problem minimize (a T x)2 − n/4 + subject to 1T x ≥ 1,
n
min{xi2 , 1/4} + min{(xi − 1)2 , 1/4}
i=1
which has the form (1), is zero if and only if xi ∈ {0, 1}, at least one of xi = 1, and a T x = 0; in other words, the set {ai | xi = 1} sums to zero. Since the subset sum problem can be reduced to an instance of (1), we conclude that in general our problem is at least as hard as difficult problems like the subset sum problem [11]. Global solution. There is a simple (exhaustive
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