Duality of nonconvex optimization with positively homogeneous functions

Abstract

We consider an optimization problem with positively homogeneous functions in its objective and constraint functions. Examples of such positively homogeneous functions include the absolute value function and the p-norm function, where p is a positive real number. The problem, which is not necessarily convex, extends the absolute value optimization proposed in [O. L. Mangasarian, Absolute value programming, Computational Optimization and Applications 36 (2007) pp. [43][44][45][46][47][48][49][50][51][52][53]. In this work, we propose a dual formulation that, differently from the Lagrangian dual approach, has a closed-form and some interesting properties. In particular, we discuss the relation between the Lagrangian duality and the one proposed here, and give some sufficient conditions under which these dual problems coincide. Finally, we show that some well-known problems, e.g., sum of norms optimization and the group Lasso-type optimization problems, can be reformulated as positively homogeneous optimization problems.Recently, the so-called absolute value equations (AVE) and absolute value optimization (AVO) problems have been attracted much attention. The AVE were introduced in 2004 by Rohn [21]. Basically, ifÃ,B are given matrices, andb is a given vector, one should find a vector x that satisfiesÃx +B|x| =b, where |x| is a vector whose i-th entry is the absolute value of the i-th entry of x. It is known that AVE are equivalent to the linear complementarity problems (LCP) [9,16,20], which include many real-world applications. As an extension of AVE, Mangasarian [14] proposed in 2007 the AVO problems, which have the absolute value of variables in their objective and constraint functions. More precisely, the AVO problem considered is given by+K|x| ≥p, whereÃ,B,H,K are given matrices, andc,d,b,p are vectors with appropriate dimensions. Since AVE and LCP are equivalent, the AVO include the mathematical programs with linear complementarity constraints [12], which are one of the formulations of equilibrium problems. As another application of AVO, Yamanaka and Fukushima [26] presented facility location problems. Since 2007, some methods for solving AVE have been presented in the literature. For example, Rohn [22] considered an iterative algorithm using the sign of variables for the case thatà andB are square matrices. For more generalà andB, Mangasarian [14] provided a method involving successive linearization techniques. Another methods include a concave minimization approach, given by Mangasarian [13], and Newton-type methods, proposed by Caccetta et al. [3], Mangasarian [15], and Zhang and Wei [28]. Some generalizations of AVE were also proposed. For example, Hu et al. [10] considered an AVE involving the absolute value of variables associated to the second-order cones. Miao et al. [18]investigated an AVE with the so-called circular cones. In both papers, quasi-Newton based algorithms were used.As for AVO problems, Yamanaka and Fukushima [26] proposed to use a branch-andbound technique. In the branching procedure, ...

Duality of optimization problems with gauge functions

“…Then we can adopt an indicator function of some cones as ψ i . We will later show that the same results as in [13] can be obtained even when domΨ = R n .…”

“…Therefore, these GO frameworks cannot directly handle linear conic optimization problems. More recently, Yamanaka and Yamashita [13] considered the following positively homogeneous optimization (PHO) problem:…”

“…Here, we explicitly include x ∈ domΨ in the constraints of (P PHO ). This is because we want to consider more general PHO problems than the ones used in the previous work [13], where domΨ = R n is assumed. Then we can adopt an indicator function of some cones as ψ i .…”

“…Yamanaka and Yamashita [13] proposed a closed dual formulation of the PHO, which they call positively homogeneous dual, and showed that the weak duality holds. They also investigated the relation between the positively homogeneous dual and the Lagrangian dual of (P PHO ), and proved that those problems are equivalent under some conditions.…”

“…Although the PHO problem has the above nice features, the theoretical analysis is still insufficient. In particular, the paper [13] does not discuss the strong duality and the primal recovery.…”