Session: Mixed-Integer Nonlinear Programming
Chair: Jan Kronqvist
Cluster: Interplay Between Continuous and Discrete Optimization
Talk 1: A graphical framework for global optimization of mixed-integer nonlinear programs
Speaker: Danial Davarnia
Abstract: Despite advances in mixed-integer linear and convex programming solvers, general mixed-integer nonlinear programs (MINLPs) remain difficult to solve due to the complex algebraic structures that modern solvers struggle to handle. This work presents a novel graphical framework for globally solving MINLPs using decision diagrams (DDs), which model complex structures beyond the reach of conventional techniques. Key components include graphical reformulation of constraints, convexification methods, efficient cutting planes, and a spatial branch-and-bound method with convergence guarantees. This work fills a gap in DD literature by offering a general-purpose solution method for MINLPs and demonstrates its efficacy on challenging test cases from the MINLP Library that cannot be solved by current global solvers
Talk 2: New perspectives on invexity and its algorithmic applications
Speaker: Ksenia Bestuzheva
Abstract: One of the key properties of convex problems is that every stationary point is a global optimum, and nonlinear programming algorithms that converge to local optima are thus guaranteed to find the global optimum. However, some nonconvex problems possess the same property. This observation has motivated research into generalizations of convexity. This talk proposes a new generalization which we refer to as optima-invexity: the property that only one connected set of optimal solutions exists. We state conditions for optima-invexity of unconstrained problems and discuss structures that are promising for practical use, and outline algorithmic applications of these structures.
Talk 3: To be updated: Mixed-integer SDP
Speaker: Jan Kronqvist
Abstract: To be updated: We consider different methods for generating cuts and solving mixed-integer semidefinite programming (MISDP) instances within an outer approximation framework. In fact, the main components of the classical outer approximation algorithm for convex mixed-integer nonlinear programming can easily be tailored towards MISDP such that similar convergence properties are obtained. We propose some new methods for generating cuts that have desirable theoretical and computational properties, and we present a numerical comparison.
