Duality of nonconvex optimization with positively homogeneous functions
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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 Mangasarian (Comput Optim Appl 36:43–53, 2007). 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.
KeywordsPositively homogeneous functions Duality Nonconvex optimization
The authors are grateful to Prof. Ellen. H. Fukuda for helpful comments and suggestions. This work was supported in part by a Grant-in-Aid for Scientific Research (C) (17K00032) from Japan Society for the Promotion of Science.
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