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

2021 ◽  
Author(s):  
Martin Bichler ◽  
Stefan Waldherr
Keyword(s):  
Computational Complexity ◽  
Core Stability ◽  
Mixed Integer ◽  
Polynomial Hierarchy ◽  
Budget Constraints ◽  
Linear Programs ◽  
Market Outcomes ◽  
Constraint Generation ◽  
Price Functions ◽  
Realistic Problem

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.

scholarly journals Bi-objective design-for-control of water distribution networks with global bounds

2021 ◽  
Author(s):  
Aly-Joy Ulusoy ◽  
Filippo Pecci ◽  
Ivan Stoianov
Keyword(s):  
Water Distribution ◽  
Distribution Networks ◽  
Water Distribution Networks ◽  
Mixed Integer ◽  
Global Optimality ◽  
Linear Programs ◽  
Design For Control ◽  
Non Linear ◽  
Feasible Solutions

AbstractThis manuscript investigates the design-for-control (DfC) problem of minimizing pressure induced leakage and maximizing resilience in existing water distribution networks. The problem consists in simultaneously selecting locations for the installation of new valves and/or pipes, and optimizing valve control settings. This results in a challenging optimization problem belonging to the class of non-convex bi-objective mixed-integer non-linear programs (BOMINLP). In this manuscript, we propose and investigate a method to approximate the non-dominated set of the DfC problem with guarantees of global non-dominance. The BOMINLP is first scalarized using the method of $$\epsilon $$ ϵ -constraints. Feasible solutions with global optimality bounds are then computed for the resulting sequence of single-objective mixed-integer non-linear programs, using a tailored spatial branch-and-bound (sBB) method. In particular, we propose an equivalent reformulation of the non-linear resilience objective function to enable the computation of global optimality bounds. We show that our approach returns a set of potentially non-dominated solutions along with guarantees of their non-dominance in the form of a superset of the true non-dominated set of the BOMINLP. Finally, we evaluate the method on two case study networks and show that the tailored sBB method outperforms state-of-the-art global optimization solvers.

scholarly journals Distributionally robust mixed integer linear programs: Persistency models with applications

2014 ◽  
Vol 233 (3) ◽  
pp. 459-473 ◽  
Author(s):  
Xiaobo Li ◽  
Karthik Natarajan ◽  
Chung-Piaw Teo ◽  
Zhichao Zheng
Keyword(s):  
Mixed Integer ◽  
Linear Programs ◽  
Integer Linear Programs ◽  
Distributionally Robust

Support Set Invariancy for Interval Bimatrix Games

2019 ◽  
Vol 27 (02) ◽  
pp. 225-237
Author(s):  
Milan Hladík
Keyword(s):  
Mixed Integer ◽  
Necessary Condition ◽  
Linear Programs ◽  
Bimatrix Games ◽  
Bimatrix Game ◽  
Integer Programs ◽  
Mixed Integer Programs ◽  
Support Set ◽  
First Case ◽  
Pure Strategies

Traditionally, game theory problems were considered for exact data, and the decisions were based on known payoffs. However, this assumption is rarely true in practice. Uncertainty in measurements and imprecise information must be taken into account. The interval-based approach for handling such uncertainties assumes that one has lower and upper bounds on payoffs. In this paper, interval bimatrix games are studied. Especially, we focus on three kinds of support set invariancy. Support of a mixed strategy consists of that pure strategies having positive probabilities. Given an interval-valued bimatrix game and supports for both players, the question states as follows: Does every bimatrix game instance have an equilibrium with the prescribed support? The other two kinds of invariancies are slight modifications: Has every bimatrix game instance an equilibrium being a subset/superset of the prescribed support? It is computationally difficult to answer these questions: the first case costs solving a large number of linear programs or mixed integer programs. For the remaining two cases a sufficient condition and a necessary condition are proposed, respectively.

An optimality cut for mixed integer linear programs

1999 ◽  
Vol 119 (3) ◽  
pp. 671-677
Author(s):  
Gilbert Laporte ◽  
Frédéric Semet
Keyword(s):  
Mixed Integer ◽  
Linear Programs ◽  
Integer Linear Programs

scholarly journals COMPARISON AND OPTIMIZATION OF CORDON AND AREA PRICINGS FOR MANAGING TRAVEL DEMAND

Transport ◽  
2014 ◽  
Vol 29 (3) ◽  
pp. 248-259 ◽  
Author(s):  
Lihui Zhang ◽  
Huiyuan Liu ◽  
Daniel (Jian) Sun
Keyword(s):  
Sensitivity Analysis ◽  
Travel Demand ◽  
Central Business District ◽  
Mixed Integer ◽  
Linear Programs ◽  
Pricing Strategies ◽  
Unified Framework ◽  
Travel Demand Management ◽  
Nonlinear Programs ◽  
Business District

This paper analyses both the cordon and area pricings from the perspective of travel demand management. Sensitivity analysis of various performance measures with respect to the toll rate and demand elastic parameter is performed on a virtual grid network. The analysis shows that cordon pricing mainly affects those trips with origins outside of the Central Business District and destinations inside, while area pricing imposes additional cost on the trips with either origins or destinations in the Central Business District. Though both pricing strategies are able to alleviate traffic congestion in the charging area, area pricing seems more effective, however, area pricing owns the risk to detour too much traffic and thus cause severe congestion to the network outside of the Central Business District. Following the sensitivity analysis, a unified framework is proposed to optimize the designs of the both pricing strategies, which is flexible to account for various practical concerns. The optimization models are formulated as mixed-integer nonlinear programs with complementarity constraints, and the solution procedure is composed of solving a series of nonlinear programs and mixed-integer linear programs. Results from the numerical examples are in line with the findings in the sensitivity analysis. Under the specific network settings, cordon pricing achieves the best system performance when the toll rate reaches the maximum allowed, while area pricing finds the optimal design scheme when the toll rate equals half of the maximum allowed.

scholarly journals Analyzing Infeasible Mixed-Integer and Integer Linear Programs

1999 ◽  
Vol 11 (1) ◽  
pp. 63-77 ◽  
Author(s):  
Olivier Guieu ◽  
John W. Chinneck
Keyword(s):  
Mixed Integer ◽  
Linear Programs ◽  
Integer Linear Programs

scholarly journals Simplified group activity selection with group size constraints

2021 ◽  
Author(s):  
Andreas Darmann ◽  
Janosch Döcker ◽  
Britta Dorn ◽  
Sebastian Schneckenburger
Keyword(s):  
Computational Complexity ◽  
Simple Model ◽  
Lower Bounds ◽  
Group Activity ◽  
Core Stability ◽  
Upper And Lower Bounds ◽  
Size Constraints ◽  
Broad Variety ◽  
Special Case ◽  
Additional Constraints

AbstractSeveral real-world situations can be represented in terms of agents that have preferences over activities in which they may participate. Often, the agents can take part in at most one activity (for instance, since these take place simultaneously), and there are additional constraints on the number of agents that can participate in an activity. In such a setting, we consider the task of assigning agents to activities in a reasonable way. We introduce the simplified group activity selection problem providing a general yet simple model for a broad variety of settings, and start investigating its special case where upper and lower bounds of the groups have to be taken into account. We apply different solution concepts such as envy-freeness and core stability to our setting and provide a computational complexity study for the problem of finding such solutions.

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