Session: Nonmonontone inclusions and nonconvex optimization
Chair: Dimitri Papadimitriou
Cluster: Nonlinear Optimization
Talk 1: Proximal iterations for solving nonmonotone inclusions and applications
Speaker: Dimitri Papadimitriou
Abstract: In a real Hilbert space H, the monotone inclusion problem aims at finding a zero of a set-valued maximally monotone operator A. The term "warped proximal iteration" was recently introduced as generalization of the proximal point iterative algorithm for finding a zero point of a maximally monotone operator A acting on the space H. Nevertheless, the maximal monotonicity of the operator A restricts the applicability of this algorithm to the class of convex optimization problems as well as operator splitting methods for composite monotone inclusions. The solving of general nonmonotone inclusion, i.e., the inclusion where the operator A is nonmonotone, remains an open and challenging research problem. For this purpose, the notion of r-(co)hypomonotonicity has been introduced to guarantee the convergence of the generated sequence. From this perspective, the first objective of this paper is to extend the definition of r-hypomotonicity. The second is to determine the convergence property of the warped proximal iteration as well as its various applications for the solving of (constrained) nonconvex optimization problems. In particular, we place our attention to the finding of Karush Kuhn Tucker (KKT) points for a class of nonconvex quadratic programming (QP) problems with equality constraints. Numerical results illustrate the performance of our method against state-of-the-art QP solvers.
Talk 2: Qualitative Analysis and Boosted DCA for Generalized Multi-Source Weber Problems
Speaker: Tuyen Tran
Abstract: In this talk, we first investigate fundamental qualitative properties of the generalized multi-source Weber problem formulated using the Minkowski gauge function. Then, we apply Nesterov's smoothing and the adaptive Boosted Difference of Convex functions Algorithm (BDCA) to solve both the unconstrained and constrained versions of the generalized multi-source Weber problems. These algorithms are tested in Matlab with real and artificial data sets. We conduct a comprehensive evaluation of the new algorithms and provide insights into its efficiency.
Talk 3: A Primal-Dual Proximal Outer Approximation algorithm
Speaker: Vu Bang
Abstract: In this talk we propose a Proximal Outer Approximation (PrOA) algorithm for solving Mixed Integer Quadratically Constrained Quadratic Programs (MI-QCQP). The proposed sequential method works as follows: starting from the integer-feasible point obtained by solving the continuous relaxation of the original problem, at each iteration, it first solves by means of a primal-dual proximal algorithm an NLP problem to obtain a constraint-feasible pair that is (also) used to compute an upper-bound on the optimal value, it then obtains a new integer-feasible pair by solving an MIQP through iteratively improving first-order approximations of the nonlinear constraints and computes a lower bound using this new pair. The integer-feasible point obtained is then used to solve the NLP problem of the next iteration. Convergence and computational properties are analyzed and compared together with numerical executions on classical QCQP's such as the QKP, the congestion-constrained FLP, and the generalized multi-source Weber problem.
