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, ...

In recent years, the absolute value equation (AVE) has attracted a growing attention. The absolute value program (AVP) is an extension of AVE, which contains absolute values of variables in its objective function and constraints. In this paper, we propose an algorithm for the AVP, which is based on the branch-and-bound method. In the branching procedure, we generate two subproblems by restricting the sign of a variable to be nonnegative or nonpositive. In the bounding procedure, we utilize the duality results for AVP. Furthermore, we carry out numerical experiments for nonconvex multi-facility location problems to show the validity of the proposed algorithm.

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Recently, Yamanaka and Yamashita proposed the so-called positively homogeneous optimization problem, which includes many important problems, such as the absolute-value and the gauge optimizations. They presented a closed dual formulation for the problem, and showed the weak duality and the equivalence to the Lagrangian dual under some conditions. In this work, we focus on a special positively homogeneous optimization problem, whose objective function and constraints consist of some gauge and linear functions. We prove not only the weak duality but also the strong one. We also study necessary and sufficient optimality conditions associated to the problem. Moreover, we give sufficient conditions under which we can recover a primal solution from a Karush-Kuhn-Tucker point of the dual formulation. Finally, we discuss how to extend the above results to general convex optimization problems by considering the so-called perspective functions.

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