Session: Advanced Techniques in Global Optimization
Chair: Sonia Cafieri
Cluster: Global Optimization
Talk 1: Towards McCormick Envelopes for Mixed-integer PDE-constrained Optimization
Speaker: Sven Leyffer
Abstract: McCormick envelopes are a well-known standard tool for deriving convex relaxations that can, for example, be used in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the McCormick nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. The resulting approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We prove that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero.
Talk 2: An Adaptive Proximal ADMM for Nonconvex Composite Programs
Speaker: Leandro Maia
Abstract: We developed an adaptive Proximal Alternating Direction Method of Multipliers (P-ADMM) for solving linearly-constrained, weakly convex, composite optimization problems. This method is adaptive to all problem parameters, including smoothness and weak convexity constants. Without any rank assumptions on the constraint matrices, it is shown that the adaptive P-ADMM obtains an approximate first-order stationary point of the constrained problem in a number of iterations that matches the state-of-the-art complexity for the class of P-ADMMs.
Talk 3: On the computation of upper bounds for some structured nonlinear minimization problems
Speaker: Sonia Cafieri
Abstract: We focus on nonlinear minimization problems that share a common structure, namely a combinatorial aspect that comes only from disjunctive constraints. For this kind of constraints, usually addressed introducing auxiliary binary variables and then handling mixed-integer formulations, a continuous-optimization alternative named 'continuous quadrant penalty formulation' was recently introduced. It is based on the introduction of a smooth piecewice-quadratic penalty function, and yields a continuous nonconvex problem. We build on this problem, to derive an efficient computation of upper bounds to be used within Branch-and-Bound-based approaches for problems arising in different domains of application. Numerical experiences show the effectiveness of these approaches at speeding up the computational convergence.
