Significant early contributions were also given in the field of unconstrained optimization, with the introduction of non monotone line searches, non monotone globalization strategies and convergent derivative-free line search techniques. The Continuous Optimization group later expanded into an active and highly valued optimization research team with a wide range of interests.
The following areas are object of current research.
• Exact penalty and augmented Lagrangian methods, still constituting the founding block of many optimization methods and a springboard for many of the studies of the group.
• Non-monotone methods and decomposition techniques for the solution of difficult large-scale nonlinear optimization problems and nonlinear equations.
• Preconditioning Newton-Krylov and Nonlinear Conjugate Gradient methods in nonconvex large scale optimization, which is an important tool for efficiently solving large difficult problems.
• Derivative-free algorithms, of special interest in many engineering applications where even the calculation of function values is problematic and very time-consuming.
• Global optimization, which is an essential tool for solving problems where local non-global solutions may be meaningless.
• Semidefinite programming, which plays an essential role in the development of efficient algorithms for solving relaxations of non-convex and integer problems.
• Finite dimensional variational inequalities and complementarity problems, which often arise in modeling a wide array of real-world problems where competition is involved.
• Generalized Nash equilibrium problems, which are emerging as a winning way of looking at several classical and non-classical engineering problems.
• Training methods for neural networks and support vector machines, for constructing surrogate models of complex systems from sparse data through learning techniques.
• Mixed Integer Nonlinear Programming (MINLP) problems that combine combinatorial aspects with nonlinearities.
• Algorithms for bilevel optimization problems in machine learning using gradient-based and implicit function approaches.
• Stochastic optimization methods under heavy-tailed noise and with guarantees in high probabilities. Stochastic and randomized proximal spitting methods.
The Continuous Optimization group interacts intensively with many other research groups, both in the academic and industrial world, in an ongoing cross-fertilization process. This process led to several innovative applications in such different fields as:
• Design of electro-mechanic devices.
• Development of electromagnetic diagnostic equipments.
• Power allocation in TLC.
• Shape optimization in ship design.
• Multiobjective optimization of nanoelectronic devices.
• Optimization of ship itineraries for a cruise fleet.
