Gerardo A. Pérez-Valdés

Bilevel Programming and Applications

A great amount of new applied problems in the area of energy networks has recently arisen that can be efficiently solved only as mixed-integer bilevel programs. Among them are the natural gas cash-out problem, the deregulated electricity market equilibrium problem, biofuel problems, a problem of designing coupled energy carrier networks, and so forth, if we mention only part of such applications. Bilevel models to describe migration processes are also in the list of the most popular new themes of bilevel programming, as well as allocation, information protection, and cybersecurity problems. This survey provides a comprehensive review of some of the above-mentioned new areas including both theoretical and applied results.

Natural gas cash-out problem: Bilevel stochastic optimization approach

A stochastic formulation of the natural gas cash-out problem is given in a form of a bilevel multi-stage stochastic programming model with recourse. After reducing the original formulation to a bilevel linear problem, a stochastic scenario tree is defined by its node events, and time series forecasting is used to produce stochastic values for data of natural gas price and demand. Numerical experiments were run to compare the stochastic solution with the perfect information solution and the expected value solutions.

Natural gas bilevel cash-out problem: Convergence of a penalty function method

This paper studies a special bi-level programming problem that arises from the dealings of a Natural Gas Shipping Company and the Pipeline Operator, with facilities of the latter used by the former. Because of the business relationships between these two actors, the timing and objectives of their decision-making process are different and sometimes even opposed. In order to model that, bi-level programming was traditionally used in previous works. Later, the problem was expanded and theoretically studied to facilitate its solution; this included extension of the upper level objective function, linear reformulation, heuristic approaches, and branch-and-bound techniques. In this paper, we present a linear programming reformulation of the latest version of the model, which is significantly faster to solve when implemented computationally. More importantly, this new formulation makes it easier to analyze the problem theoretically, allowing us to draw some conclusions about the nature of the solution of the modified problem. Numerical results concerning the running time, convergence, and optimal values, are presented and compared to previous reports, showing a significant improvement in speed without actual sacrifice of the solutions quality.

A parallelised distributed implementation of a Branch and Fix Coordination algorithm

Branch and Fix Coordination is an algorithm intended to solve large scale multi-stage stochastic mixed integer problems, based on the particular structure of such problems, so that they can be broken down into smaller subproblems. With this in mind, it is possible to use distributed computation techniques to solve the several subproblems in a parallel way, almost independently. To guarantee non-anticipativity in the global solution, the values of the integer variables in the subproblems are coordinated by a master thread. Scenario ‘clusters’ lend themselves particularly well to parallelisation, allowing us to solve some problems noticeably faster. Thanks to the decomposition into smaller subproblems, we can also attempt to solve otherwise intractable instances. In this work, we present details on the computational implementation of the Branch and Fix Coordination algorithm.

US Natural Gas Market Classification Using Pooled Regression

Natural gas marketing has considerably evolved since the early 1990s, when a set of liberalizing rules were passed in both the United States and the European Union that eliminated state-driven regulations in favor of open energy markets. These new rules changed many things in the business of energetics, and therefore new research opportunities arose. Econometric studies about natural gas emerged as an important area of study since natural gas may now be sold and traded in a number of stock markets, each one responding to potentially different behavioral drives. In this work, we present a method to differentiate sets of time series based on a regression model relating price, consumption, supply, and other factors. Our objective is to develop a method to classify different areas, regions, or states into groups or classes that share similar regression parameters. Once obtained, these groups may be used to make assumptions about corresponding natural gas prices in further studies.

Reduction of Dimension of the Upper Level Problem in a Bilevel Programming Model Part 1

The paper deals with a problem of reducing dimension of the upper level problem in a bilevel programming model. In order to diminish the number of variables governed by the leader at the upper level, we create the second follower supplied with the objective function coinciding with that of the leader and pass part of the upper level variables to the lower level to be governed but the second follower. The lower level problem is also modified and becomes a Nash equilibrium problem solved by the original and the new followers. We look for conditions that guarantee that the modified and the original bilevel programming problems share at least one optimal solution.

Reduction of Bilevel Programming to a Single Level Problem

Using the Karush-Kuhn-Tucker conditions, generalized equations describing necessary optimality conditions or the optimal value function of the lower level problem, the bilevel optimization problem can be transformed into a single-level optimization problem. Two of these transformations are fully equivalent to the bilevel problem, the MPEC is not. Using these transformations, necessary conditions for local optimal solutions can be formulated: In the case of a strongly stable lower level optimal solution using its directional derivative, using partial calmness in the optimal value function transformation, and applying variational analysis for KKT transformations explicitly using Lagrange multipliers or not. Solution algorithms are formulated and investigated for all reductions.

Convex Bilevel Programs

The task to find a best point within the set of optimal solutions of a convex optimization problem is called simple bilevel optimization problem. In general, a necessary optimality condition for a convex simple bilevel problem does not need to be sufficient. An adapted necessary and sufficient optimality condition is derived using tools from cone-convex optimization and a gradient type descent method is suggested which combines the use of a convex combination of both objective functions and projection onto the feasible set. In the second section, a similar algorithm is used to find a best point within the solutions of a variational inequality.

Applications to Natural Gas Cash-Out Problem

The nature of natural gas implies that imbalances occur between the declared and really extracted amount of gas in a pipeline system. The transmit system operator penalizes the gas shipping company for these imbalances. The task of minimizing the resulting cash flow is modeled as bilevel optimization problem with one Boolean variable in the lower level problem. We report related models, arising in applications, if the Boolean variable is shifted to the upper level problem or when stochastic data are implemented. Solution algorithms and numerical results are presented.

Linear Bilevel Optimization Problem

The linear bilevel optimization problem is considered first. For this some surprising properties are reported: What happens if a constraint on both the upper and the lower level variables is moved from the upper to the lower level problem or one constraint is added which is not active in the lower level problem at an optimal solution? What happens if a variable is added in the lower level? The bilevel optimization problem is a \(NP\)- hard optimization problem but conditions can be formulated guaranteeing that verification of an optimal solution can be done in polynomial time. In the last part, solution algorithms for the linear bilevel optimization problem are formulated either using regions of stability for solutions of the lower level problem or the optimal value reformulation of the bilevel problem.