Optimization applications combine technology and expertise from many different areas, including model-building, algorithms, and data-handling. Often, the gathering, pre/post-processing, and visualization of the data is done by a diverse organization-spanning group that shares a common bond: their skill in and appreciation for Python and the vast array of available packages it provides. For this reason, GAMS offers a new comfortable way to integrate with Python on the data-handling and modeling side. In this talk, we will explore the benefits of our Python library GAMSPy.

We are investigating the modeling and design of algorithms for models with multiple optimization problems.
This structure is present in many problem classes like bilevel/MPEC/multilevel programming or Nash equilibrium problems or whenever value functions present in the mappings of an optimization problem, like in multistage stochastic programming with coherent risk measures. Here, we tackle the challenge of modeling setups with any combination of the above structures.
For instance, decentralized energy markets with part of the problem data being subjected to uncertainty can feature a Nash equilibrium problem where market participants have a nonsmooth objective function due to the presence of a coherent risk measure.

For such complex but structured setups, we propose a modeling framework based on a directed acyclic graph (DAG) structure.
The nodes are of two types: the first one subsumes parametric optimization problems and variational inequalities, while the second one indicates a Nash behavior between its children nodes. Edges between two optimization problems specifies their interaction: either a hierarchical one (bilevel/MPEC) or the value function of the child appears in the parent problem.

This DAG structure is leveraged by model transformations to transform part or all of the problem in a form amenable to computation by existing solvers. When tackling some discrete stochastic programs, the reformulated problems do then belong to a classical model type and high-performance solvers for discrete optimization problem can be used to compute a solution.
The DAG is also useful in the design of decomposition algorithms for solving complex instances. An implementation of these ideas is present in ReSHOP, a reformulation solver for hierarchical optimization problems.

We take it for granted that an optimization package accepts both minimization and maximization problems, recognizes them as equivalent, and converts all minimizations to maximizations (or vice-versa) before solving. This is only the very simplest example of the many conversions carried out routinely by large-scale optimization software. The range of expressions recognized by modeling languages and solvers has been steadily extended in ways that make optimization models easier to describe, validate, and maintain — but that make conversion possibilities ever more numerous and complex.

Continuing this trend, automatic conversions have been a central feature in the design of a new solver interface framework for the AMPL modeling language. This presentation describes a range of challenges that have been faced in detecting formulations that solvers can handle, and in implementing conversions to forms that solvers require. Examples combining a variety of discrete and nonlinear expressions lead to some general recommendations for design and implementation of automated conversions.

We present a mechanism for describing and solving collections of optimization problems that are linked by equilibrium conditions. The general framework of MOPEC (multiple optimization problems with equilibrium constraints) captures many example applications that involve independent decisions coupled by shared resources. Included in this class are classical models such as complementarity, general equilibria, and agent based formulations arising from Nash Games.

Stochastic equilibria can be used to model many data driven applications, including many dynamic models of competition. In this model, players solve risk averse optimization problems that are coupled by a shared scenario tree. Players are linked by equilibrium conditions at nodes of that tree. We outline several algorithms and investigate their computational efficiency. Applications to energy planning and their interactions with environmental concerns will be outlined, and we will demonstrate how these problems can be implemented in modeling systems.