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Stream: Analysis and engineering of optimization algorithms

Chair: Juan Enrique Martínez-Legaz

1. A general nonconvex multiduality principle
   Juan Enrique Martínez-Legaz, Francesca Bonenti, Rossana Riccardi
   
   
   

   We present an arbitrary collection of mutually dual nonconvex optimization problems, as well as a characterization of their global optimal solutions. As immediate consequences of our general multiduality principle, we obtain Toland-Singer duality theorem and an analogous result involving generalized perspective functions.
   
   

2. On Some Methods to Find Stationary Points of Quasidifferentiable Functions with DC-Duality Approach
   Didem Tozkan, Mahide Kucuk, Yalcin Kucuk
   
   
   

   In this study, we introduce some methods to find stationary points of a quasidifferentiable optimization problem (QDP), i.e., the directional derivative of the objective is a DCH-function. Firstly, we obtain some optimality conditions for DCH-functions by using DC-duality approach. As a result, we give some methods to find stationary points for (QDP) considering the problem (P') that minimizes the directional derivative of the original objective function. We present some illustrative examples about given methods.
   
   

3. A Reformulation of Sparse Optimization Problems using Complementarity-type Constraints
   Alexandra Schwartz, Christian Kanzow, Oleg Burdakov
   
   
   

   We consider sparse optimization problems, i.e. mathematical programs where the objective is not only to minimize a given function but also the number of nonzero elements in the solution vector. Possible applications are compressed sensing, portfolio optimization and feature selection. In this talk, we present a continuous reformulation of the noncontinuous sparsity term in the objective function using a complementarity-type constraint. We discuss the relation between the original and the reformulated problem, provide suitable optimality conditions and provide preliminary numerical results.
   
   

4. A continuous DC programming approach to nonlinear mixed integer programs without integrality gaps
   Takayuki Okuno, Yoshiko Ikebe
   
   
   

   In this talk, we consider a mixed integer nonconvex program (MINP). In particular, we restrict ourselves to the MINP whose objective function is a dc function, that is, a function that can be represented as the difference of two convex functions. Based on a new technique proposed by T. Maehara, et al.(2015), we transform the MINP to a certain equivalent continuous DC program. For solving it, we propose a proximal point type DC algorithm. Under several mild assumptions, we prove that the sequence generated by the proposed method converges to some stationary point of the MINP.