176 research outputs found
Solving Mathematical Programs with Equilibrium Constraints as Nonlinear Programming: A New Framework
We present a new framework for the solution of mathematical programs with
equilibrium constraints (MPECs). In this algorithmic framework, an MPECs is
viewed as a concentration of an unconstrained optimization which minimizes the
complementarity measure and a nonlinear programming with general constraints. A
strategy generalizing ideas of Byrd-Omojokun's trust region method is used to
compute steps. By penalizing the tangential constraints into the objective
function, we circumvent the problem of not satisfying MFCQ. A trust-funnel-like
strategy is used to balance the improvements on feasibility and optimality. We
show that, under MPEC-MFCQ, if the algorithm does not terminate in finite
steps, then at least one accumulation point of the iterates sequence is an
S-stationary point
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Using EPECs to model bilevel games in restructured electricity markets with locational prices
CWPE0619 (EPRG0602) Xinmin Hu and Daniel Ralph (Feb 2006) Using EPECs to model bilevel games in restructured electricity markets with locational prices We study a bilevel noncooperative game-theoretic model of electricity markets with locational marginal prices. Each player faces a bilevel optimization problem that we remodel as a mathematical program with equilibrium constraints, MPEC. This gives an EPEC, equilibrium problem with equilibrium constraints. We establish sufficient conditions for existence of pure strategy Nash equilibria for this class of bilevel games and give some applications. We show by examples the effect of network transmission limits, i.e. congestion, on existence of equilibria. Then we study, for more general EPECs, the weaker pure strategy concepts of local Nash and Nash stationary equilibria. We model the latter via complementarity problems, CPs. Finally, we present numerical examples of methods that attempt to find local Nash or Nash stationary equilibria of randomly generated electricity market games. The CP solver PATH is found to be rather effective in this context
Dynamic Congestion and Tolls with Mobile Source Emission
This paper proposes a dynamic congestion pricing model that takes into
account mobile source emissions. We consider a tollable vehicular network where
the users selfishly minimize their own travel costs, including travel time,
early/late arrival penalties and tolls. On top of that, we assume that part of
the network can be tolled by a central authority, whose objective is to
minimize both total travel costs of road users and total emission on a
network-wide level. The model is formulated as a mathematical program with
equilibrium constraints (MPEC) problem and then reformulated as a mathematical
program with complementarity constraints (MPCC). The MPCC is solved using a
quadratic penalty-based gradient projection algorithm. A numerical study on a
toy network illustrates the effectiveness of the tolling strategy and reveals a
Braess-type paradox in the context of traffic-derived emission.Comment: 23 pages, 9 figures, 5 tables. Current version to appear in the
Proceedings of the 20th International Symposium on Transportation and Traffic
Theory, 2013, the Netherland
SOLVING TWO-LEVEL OPTIMIZATION PROBLEMS WITH APPLICATIONS TO ROBUST DESIGN AND ENERGY MARKETS
This dissertation provides efficient techniques to solve two-level optimization problems. Three specific types of problems are considered. The first problem is robust optimization, which has direct applications to engineering design. Traditionally robust optimization problems have been solved using an inner-outer structure, which can be computationally expensive. This dissertation provides a method to decompose and solve this two-level structure using a modified Benders
decomposition. This gradient-based technique is applicable to robust optimization problems with quasiconvex constraints and provides approximate solutions to problems with nonlinear constraints. The second types of two-level problems considered are mathematical and equilibrium programs with equilibrium constraints. Their two-level structure is simplified using Schur's decomposition and reformulation
schemes for absolute value functions. The resulting formulations are applicable to game theory problems in operations research and economics. The third type of two-level problem studied is discretely-constrained mixed linear complementarity problems. These are first formulated into a two-level mathematical program with equilibrium constraints and then solved using the aforementioned technique for mathematical and equilibrium programs with equilibrium constraints. The techniques for all three problems help simplify the two-level structure into one level, which helps gain numerical and application insights. The computational effort for solving these problems is greatly reduced using the techniques in this dissertation. Finally, a host of numerical examples are presented to verify the approaches. Diverse applications to economics, operations research, and engineering design motivate the relevance of the novel methods developed in this dissertation
Complementarity, not optimization, is the language of markets
Each market agent (producer or consumer) in a power market pursues its own objective, typically to maximize its own profit. As such, the specific behavior of each agent in the market is conveniently formulated as a bi-level optimization problem whose upper-level problem represents the profit seeking behavior of the agent and whose lower-level problem represents the clearing of the market. The objective function and the constraints of this bi-level problem depend on the agent's own decision variables and on those of other agents as well. Understanding the outcomes of the market requires considering and solving jointly the interrelated bi-level problems of all market agents, which is beyond the purview of optimization. Solving jointly a set of bi-level (or single-level) optimization problems that are interrelated is the purview of complementarity. In this paper and in the context of power markets, we review complementarity using a tutorial approach
A Coevolutionary Particle Swarm Algorithm for Bi-Level Variational Inequalities: Applications to Competition in Highway Transportation Networks
A climate of increasing deregulation in traditional highway transportation,
where the private sector has an expanded role in the provision of traditional
transportation services, provides a background for practical policy issues to be investigated.
One of the key issues of interest, and the focus of this chapter, would
be the equilibrium decision variables offered by participants in this market. By assuming
that the private sector participants play a Nash game, the above problem can
be described as a Bi-Level Variational Inequality (BLVI). Our problem differs from
the classical Cournot-Nash game because each and every player’s actions is constrained
by another variational inequality describing the equilibrium route choice of
users on the network. In this chapter, we discuss this BLVI and suggest a heuristic
coevolutionary particle swarm algorithm for its resolution. Our proposed algorithm
is subsequently tested on example problems drawn from the literature. The numerical
experiments suggest that the proposed algorithm is a viable solution method for
this problem
Vulnerability Assessment of Large-scale Power Systems to False Data Injection Attacks
This paper studies the vulnerability of large-scale power systems to false
data injection (FDI) attacks through their physical consequences. Prior work
has shown that an attacker-defender bi-level linear program (ADBLP) can be used
to determine the worst-case consequences of FDI attacks aiming to maximize the
physical power flow on a target line. This ADBLP can be transformed into a
single-level mixed-integer linear program, but it is hard to solve on large
power systems due to numerical difficulties. In this paper, four
computationally efficient algorithms are presented to solve the attack
optimization problem on large power systems. These algorithms are applied on
the IEEE 118-bus system and the Polish system with 2383 buses to conduct
vulnerability assessments, and they provide feasible attacks that cause line
overflows, as well as upper bounds on the maximal power flow resulting from any
attack.Comment: 6 pages, 5 figure
Dynamic Modelling and Optimisation of Large-Scale Cryogenic Separation Processes
In this work, the open loop dynamic optimisation of a large-scale natural gas processing plant is performed. A rigorous differential-algebraic equation (DAE) model has been formulated to represent main plant units, such as shell and tube heat exchangers, highpressure separator and demethanizing column. In the shell and tube heat exchangers, the hot stream partially condenses and equations to consider the partial condensation of the fluids have been included. A rigorous index one model for the demethanizing column has been developed. The DAE optimisation problem is solved with a simultaneous approach, in which both state and control variables are discretised and the original DAE optimisation model is transformed into a large-scale nonlinear problem (NLP), which is solved using Sequential Quadratic Programming (SQP) methods. Optimal profiles have been obtained for main operating variables to achieve an enhanced product recovery.Fil: Rodriguez, Mariela Alejandra. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas. Centro CientĂfico TecnolĂłgico Conicet - BahĂa Blanca. Planta Piloto de IngenierĂa QuĂmica. Universidad Nacional del Sur. Planta Piloto de IngenierĂa QuĂmica; ArgentinaFil: Bandoni, Jose Alberto. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas. Centro CientĂfico TecnolĂłgico Conicet - BahĂa Blanca. Planta Piloto de IngenierĂa QuĂmica. Universidad Nacional del Sur. Planta Piloto de IngenierĂa QuĂmica; ArgentinaFil: DĂaz, MarĂa Soledad. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas. Centro CientĂfico TecnolĂłgico Conicet - BahĂa Blanca. Planta Piloto de IngenierĂa QuĂmica. Universidad Nacional del Sur. Planta Piloto de IngenierĂa QuĂmica; Argentin
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