Core Pricing in Combinatorial Exchanges with Financially Constrained Buyers: Computational Hardness and Algorithmic Solutions

The computation of market equilibria is a fundamental and practically relevant problem. Although we know the computational complexity and the types of price functions necessary for combinatorial exchanges with quasilinear preferences, the respective literature does not consider financially constrained buyers. We show that computing market outcomes that respect budget constraints but are core stable is a problem in the second level of the polynomial hierarchy. Problems in this complexity class are rare, but ignoring budget constraints can lead to significant efficiency losses and instability. We introduce mixed integer bilevel linear programs (MIBLP) to compute core-stable market outcomes and provide effective column and constraint generation algorithms to solve these problems. Although full core stability quickly becomes intractable, we show that realistic problem sizes can actually be solved if the designer limits attention to deviations of small coalitions. This n-coalition stability is a practical approach to tame the computational complexity of the general problem and at the same time provides a reasonable level of stability for markets in the field where buyers have budget constraints.

Multi-Period Fast Robust Optimization for Partial Distributed Generators (DGs) Providing Ancillary Services

Distributed generators providing auxiliary service are an important means of guaranteeing the safe and economic operation of a distribution system. In this paper, considering an energy storage system (ESS), switchable capacitor reactor (SCR), step voltage regulator (SVR), and a static VAR compensator (SVC), a two-stage multi-period hybrid integer second-order cone programming (SOCP) robust model with partial DGs providing auxiliary service is developed. If the conic relaxation is not exact, a sequential SOCP is formulated using convex–concave procedure (CCP) and cuts, which can be quickly solved. Moreover, the exact solution of the original problem can be recovered. Furthermore, in view of the shortcomings of the large computer storage capacity and slow computational rate for the column and constraint generation (CCG) method, a method direct iteratively solving the master and sub-problem is proposed. Increases to variables and constraints to solve the master problem are not needed. For the sub-problem, only the model of each single time period needs to be solved. Then, their objective function values are accumulated, and the worst scenarios of each time period are concatenated. As an outcome, a large amount of storage memory is saved and the computational efficiency is greatly enhanced. The capability of the proposed method is validated with three simulation cases.

Optimized location of light sources to cover a rectangular region

The problem we considered was proposed by an industrial partner. The aim is to locate light sources around a rectangular field such that the areas these illuminate cover the whole field. We assume these illuminated areas to be rectangular as well, parallel to the field. Covering an area with multiple lights is allowed. There are several types of light sources, priced differently with different sizes of their illuminated area. We aim to minimize the cost of the cover. We propose a constraint generation approach for solving this covering problem. We formulate a MIP model to locate the light sources such that a finite number of predetermined points have to be covered. The result does not necessarily solve the original problem, i.e. it does not cover the whole field. Therefore, a constraint generation model is built to calculate a non-covered point such that the first model has to improve its previous solution to cover this new point as well. If no uncovered point is found, the result is an optimal covering, thus we stop. We have also designed some set of additional constraints to exclude symmetrical solutions, to speed up the algorithm. We analyze the efficiency of the additional constraints and report some computational results in realistic settings.

Robust Facility Location Under Disruptions

Facility networks can be disrupted by, for example, power outages, poor weather conditions, or natural disasters, and the probabilities of these events may be difficult to estimate. This could lead to costly recourse decisions because customers cannot be served by the planned facilities. In this paper, we study a fixed-charge location problem (FLP) that considers disruption risks. We adopt a two-stage robust optimization method, by which facility location decisions are made here and now and recourse decisions to reassign customers are made after the uncertainty information on the facility availability has been revealed. We implement a column-and-constraint generation (C&CG) algorithm to solve the robust models exactly. Instead of relying on dualization or reformulation techniques to deal with the subproblem, as is common in the literature, we use a linear programming–based enumeration method that allows us to take into account a discrete uncertainty set of facility failures. This also gives the flexibility to tackle cases when the dualization technique cannot be applied to the subproblem. We further develop an approximation scheme for instances of a realistic size. Numerical experiments show that the proposed C&CG algorithm outperforms existing methods for both the robust FLP and the robust p-median problem.

A column-and-constraint generation algorithm for two-stage stochastic programming problems

This paper presents a column-and-constraint generation algorithm for two-stage stochastic programming problems. A distinctive feature of the algorithm is that it does not assume fixed recourse and as a consequence the values and dimensions of the recourse matrix can be uncertain. The proposed algorithm contains multi-cut (partial) Benders decomposition and the deterministic equivalent model as special cases and can be used to trade-off computational speed and memory requirements. The algorithm outperforms multi-cut (partial) Benders decomposition in computational time and the deterministic equivalent model in memory requirements for a maintenance location routing problem. In addition, for instances with a large number of scenarios, the algorithm outperforms the deterministic equivalent model in both computational time and memory requirements. Furthermore, we present an adaptive relative tolerance for instances for which the solution time of the master problem is the bottleneck and the slave problems can be solved relatively efficiently. The adaptive relative tolerance is large in early iterations and converges to zero for the final iteration(s) of the algorithm. The combination of this relative adaptive tolerance with the proposed algorithm decreases the computational time of our instances even further.

Semi-infinite programming for trajectory optimization with non-convex obstacles

This article presents a novel optimization method that handles collision constraints with complex, non-convex 3D geometries. The optimization problem is cast as a semi-infinite program in which each collision constraint is implicitly treated as an infinite number of numeric constraints. The approach progressively generates some of these constraints for inclusion in a finite nonlinear program. Constraint generation uses an oracle to detect points of deepest penetration, and this oracle is implemented efficiently via signed distance field (SDF) versus point cloud collision detection. This approach is applied to pose optimization and trajectory optimization for both free-flying rigid bodies and articulated robots. Experiments demonstrate performance improvements compared with optimizers that handle only convex polyhedra, and demonstrate efficient collision avoidance between non-convex CAD models and point clouds in a variety of pose and trajectory optimization settings.