Course 1: Tools for convex optimization in nonpositively-curved spaces
By Adriana Nicolae (Babeş-Bolyai University)
Geodesic metric spaces that satisfy the CAT(0) inequality, a condition of non-positive curvature, provide an appropriate framework for developing a theory of convex sets and convex functions. This short course will explore fundamental geometric properties of these spaces, which are essential for understanding a range of convex optimization problems and designing algorithms for solving them. The course will cover the following topics, among others:
- Geometry of CAT(0) spaces. Cubical complexes, Gromov’s “link” condition, length and geodesic spaces, Alexandrov angles, nonpositive curvature, examples.
- Convexity in CAT(0) spaces. Convex sets, projections, convex functions, minimizers of convex functions, barycenters, circumcenters, resolvents.
- Boundary at infinity of CAT(0) spaces. Asymptotic rays, boundary at infinity, cone topology, Busemann functions.
Course 2: Lipschitz optimization with contemporary structure: a short course
Adrian Lewis (Cornell)
Classical continuous optimization addresses smooth or convex objectives on Euclidean spaces, using (sub)gradient oracles and explicit problem structure. Contemporary problems, on the other hand, influenced in part by machine learning, may involve non-Euclidean settings - manifolds, Wasserstein spaces, or geodesic metric spaces - and oracles for unstructured Lipschitz objectives. This course explores a variety of geometric ideas underpinning contemporary algorithms on these new frontiers. We emphasize several themes.
- Convex optimization in nonpositively-curved spaces. The scope of computational optimization in settings like hyperbolic space or in CAT(0) cubical complexes like tree space.
- Lipschitz optimization in Euclidean space. Condition-based complexity analysis - growth and nonconvexity. Conservative gradient fields.
- Metric analysis of non-Euclidean optimization algorithms. Kurdyka-Lojasiewicz theory, identifiability, and active-set methods.
Course 3: Proximal Algorithms in Metric Spaces: a short course
Russell Luke (Göttingen)
Proximal algorithms, together with descent methods are the engine of machine learning and artificial intelligence, as well as many other modern applications. The most natural settings for these extremely simple algorithms in many cases are extremely delicate nonlinear metric spaces, and the corresponding mappings are usually nonconvex and often set-valued. These lectures develop the foundations of proximal splitting for nonconvex optimization, and more generally set-valued fixed point iterations in uniquely geodesic metric spaces like tree spaces, spherical patches, and probability measure spaces. The lectures will be split into three topics:
- Analysis of Sequences in Nonlinear Spaces: Metric space basics, convergence rates and complexity, and multi-valued mappings and sequences of sets.
- Elements of Variational Analysis in Metric Spaces: Convex sets, basic properties of functions, application to minimization.
- Fixed Point Theory: Regularity of multi-valued mappings, convergence, stability and rates of convergence.
Course 4: Optimization with ordering: A tour of perspective
Víctor Blanco (UGR), Stefan Nickel (KIT), Justo Puerto (US), Antonio M. Rodríguez-Chía (UCA)
Mathematical optimization models are highly dependent on the choice of the measure to be maximized or minimized in the optimization process. Most of the models in the literature, as those that arise in facility location, network design, or even in machine learning, require to measure the goodness of a solution by aggregating into a single value the separated goodness measures for the different entities involved in the problem. Classical models assume that this aggregation is performed either by the average (sum) or the worst case (max or min) behavior. These two approaches have given rise to a vast applied optimization literature in different fields. However, it has been largely recognized that many more point of views make sense in different frameworks. In this line, different aggregation measures have been proposed to derive solutions for decisions problems that allow the decision-maker to select alternatives not only focused on economically efficient returns, but also in fair allocations, robust criteria in uncertain situations, portfolio selection problems under risk, balanced criteria in supply chain management, envy free measures in social choice and distribution problems.
This lecture will be organized in three different topics:
- A general framework for optimization problems with ordering.
- The ordered median location problem on continuous, networks and discrete settings.
- Applications: network design, logistics, voting, image segmentation, portfolio selection, routing, fair decision theory, robustness…
Course 5: Mathematical Models for Vehicle Routing Problems
Juan José Salazar (ULL)
The Capacitated Vehicle Routing Problem (CVRP) is the basic version of an extensively studied family of combinatorial optimization problems which model short-haul transportation applications, where the demand of the customers is typically (much) smaller than the vehicle capacity so that several customers may be served in a vehicle route. The CVRP extends the well-known Traveling Salesperson Problem (TSP) which looks for the shortest route for a single uncapacitated vehicle visiting a set of customers. Both in the TSP and in the CVRP, travel costs are given, each customer must be visited once, and vehicles start and end at a depot. The aim of both problems is to minimize the sum of travel costs. The extension from TSP to CVRP consists in caring also about customers’ demands and vehicle capacities. Hence, the CVRP in- volves solving two combinatorial optimization problems. One problem is assigning customers to vehicles in such a way that each vehicle can feasibly load all the demand required to serve all the customers assigned to it. The other problem is determining the shortest route for each vehicle visiting its assigned customers. CVRP aims to solve both combinatorial problems simultaneously. It is worth noting that each problem individually is a challenging one (i.e., it is NP hard in complexity theory terminology).
This lecture focuses on using Mathematical Programming to solve CVRP related problems, which are crucial in Logistic today. We will present, analyze and compare several mathematical formulations in mixed integer linear programming. Some models are based on a polynomial number of variables and constraints, while other models rely on exponential numbers of variables or columns. All the models are implemented in a companion Julia code to help a reader better understanding the advantages and disadvantages of using each formulation.
19/June/2026 - 20/June/2026
Offices facilities at IMUS
International Doc-Course Workshop
By Doc-Course participants
Till the 10th of July
Offices facilities at IMUS
Supervised Research Program
By Doc-Course Professors
Each student will be assigned to a supervisor for this research period.
