Mathematical Programming Problem With Equilibrium Constraints Research Articles

Iterative Backpropagation Method for Efficient Gradient Estimation in Bilevel Network Equilibrium Optimization Problems

Network optimization or network design with an embedded traffic assignment (TA) to model user equilibrium principle, sometimes expressed as bilevel problems or mathematical programs with equilibrium constraints (MPEC), is at the heart of transportation planning and operations. For applications to large-scale multimodal networks with high dimensional decision variables, the problem is nontrivial, to say the least. General-purpose algorithms and problem-specific bilevel formulations have been proposed in the past to solve small problems for demonstration purposes. Research gap, however, exists in developing efficient solution methods for large-scale problems in both static and dynamic contexts. This paper proposes an efficient gradient estimation method called Iterative Backpropagation (IB) for network optimization problems with an embedded static TA model. IB exploits the iterative structure of the TA solution procedure and simultaneously calculates the gradients while the TA process converges. IB does not require any additional function evaluation and consequently scales very well with higher dimensions. We apply the proposed approach to origin-destination (OD) estimation, an MPEC problem, of the Hong Kong multimodal network with 49,806 decision variables, 8,797 nodes, 18,207 links, 2,684 transit routes, and 165,509 OD pairs. The calibrated model performs well in matching the link counts. Specifically, the IB-gradient based optimization technique reduces the link volume squared error by 98%, mean absolute percentage error (MAPE) from 95.29% to 21.23%, and the average GEH statistics from 24.18 to 6.09 compared with the noncalibrated case. The framework, even though applied to OD estimation in this paper, is applicable to a wide variety of optimization problems with an embedded TA model, opening up an efficient way to solve large-scale MPEC or bilevel problems. Funding: The study is supported by IVADO Postdoctoral Fellowship scheme 2021, HSBC 150th Anniversary Charity Programme HKBF17RG01, National Science Foundation of China (71890970, 71890974), General Research Fund (16212819, 16207920) of the HKSAR Government, and the Hong Kong PhD Fellowship.

Generalized invexity and mathematical programs

In this paper, using generalized convexity assumptions, we show that Mstationary condition is sufficient for global or local optimality under some mathematical programming problem with equilibrium constraints(MPEC). Further, we formulate and study Wolfe type and Mond-Weir type dual models for the MPEC, and we establish weak and strong duality theorems.

A New Commerce Operation Model for Integrated Energy System Containing the Utilization of Bio-Natural Gas

To promote the collaborative development of the bio-natural gas (BNG) industry and the integrated energy system (IES), this paper proposes a new commerce operation model considering the gas price adjustment mechanism for the IES with the utilization of bio-natural gas. The bi-level optimization model is used to simulate the clearing process within the open energy market framework, and the uncertainties of variable renewable energy output are modeled with a set of scenarios through the stochastic programming approach. In the upper-level model, the energy management center adjusts the bio-natural gas price rationally to minimize the expected total operating cost and release the price signal to the lower-level model; the lower-level model simulates the sub-markets clearing process to formulate detailed operation schemes. The bi-level model is transformed into a mathematical programming problem with equilibrium constraints (MPEC) through the Karush–Kuhn–Tucher (KKT) condition of the lower-level model, and the nonlinear model is converted into a mixed-integer linear programming problem and solved. The numerical results verified the effectiveness of the proposed model.

Reward Optimization for Content Providers With Mobile Data Subsidization: A Hierarchical Game Approach

Mobile data subsidization launched by mobile network operators is a promising business model to provide economic benefits for the mobile data market and beyond. It allows content providers to partly subsidize mobile data consumption of mobile users in exchange for displaying a certain amount of advertisements. From a content provider perspective, it is of great interest to determine the optimal strategy for offering appropriate data subsidization (reward) in order to compete against others to earn more revenue and gain higher profit. In this paper, we take a hierarchical game approach to model the reward optimization process for the content providers. To analyze the relationship between the provider and the user, we first focus on the one-to-one interaction in a single-provider single-user system, and formulate a Mathematical Program with Equilibrium Constraints (MPEC). We apply the backward induction to solve the MPEC problem and prove the existence and uniqueness of the Stackelberg equilibrium. We then formulate an Equilibrium Program with Equilibrium Constraints (EPEC) to characterize the many-to-many interactions among multiple providers and multiple users. Considering the inherent high complexity of the EPEC problem, we utilize the distributed Alternating Direction Method of Multipliers (ADMM) algorithm to obtain the optimum solutions with fast-convergence and decomposition properties of ADMM.

Optimal wind energy bidding strategies in real‐time electricity market with multi‐energy sources

This study investigates optimal wind power generator bidding strategies in the real-time electricity market. The goal is to maximise its operating profit by determining the optimal amount of wind power to bid in the real-time market. A bi-level stochastic optimisation model is proposed in which the upper-level problem is minimising the negative profit of wind power producers, while the low-level problem clears the real-time market. The uncertainties in the wind power production, thermal power, hydro power, demand, and energy storage are considered in the stochastic model. This study utilises a mathematical programming problem with equilibrium constraints (MPEC) and Karush-Kuhn-Tucker (KKT) conditions to transform the bi-level problem into an equivalent single-level mixed-integer linear problem (MILP). Case studies obtained with the IEEE RTS-24 Bus system demonstrate the effectiveness of the proposed model and the effect of scenarios on the wind power producer's profit.

On the relation between MPECs and optimization problems in abs-normal form

ABSTRACTWe show that the problem of unconstrained minimization of a function in abs-normal form is equivalent to identifying a certain stationary point of a counterpart Mathematical Program with Equilibrium Constraints (MPEC). Hence, concepts introduced for the abs-normal forms turn out to be closely related to established concepts in the theory of MPECs. We give a number of proofs of equivalence or implication for the kink qualifications LIKQ and MFKQ. We also show that the counterpart MPEC always satisfies MPEC-ACQ. We then consider non-smooth nonlinear optimization problems (NLPs) where both the objective function and the constraints are presented in the abs-normal form. We show that this extended problem class also has a counterpart MPEC problem.

Allocation of street parking facilities in a capacitated network with equilibrium constraints on drivers’ traveling and cruising for parking

This study presents a method of mathematical programs with equilibrium constraints (MPEC) to determine the optimal sites of street parking facilities in a working area where drivers travel through or cruise for parking spaces in the morning commute. The optimal sites of street parking facilities incur shortest queuing delay in the capacitated network. We first propose a variational inequality to formulate the heterogeneous quantal response equilibrium (HQRE) for parkers in non-cooperative parking game and the logit-based stochastic user equilibrium (SUE) for travelers traversing the area. The HQRE is developed under the “infinite cyclic flows” assumption, allowing all possible routes, containing the cyclic routes of all times including the infinite. Therefore boundedly rational parkers who search the same curb for more than once via cyclic cruising strategies can be modeled in the parking game. Then based on the equilibrium conditions, the MPEC was developed to minimize the total queuing delay for all drivers in the area. The radial basis function (RBF) model algorithm, one of the surrogate model-based algorithms, was proposed to iteratively solve the MPEC problem. A numerical example from a real working area is presented to illustrate the proposed MPEC problem. The numerical results show that the traffic flow increase caused by cruising for parking can be presented by our model. In addition, the appropriate allocation should avoid parking spaces at the curbs of arterial where travelers drive for traversing the area and the streets intersecting the arterial.

Sharing demand-side energy resources - A conceptual design

Motivated by the recent boom of the sharing economy, this paper presents a scheme of sharing demand-side energy resources (DERs) among multiple prosumers. Successful sharing must achieve enhanced utilization efficiency of DERs and, in the mean time, ensure voluntary participation of prosumers and a sharing-enabling aggregator. It is also desirable to incentivize the adoption of DERs. To fulfill these goals, we formulate a mathematical program with equilibrium constraints (MPEC) for DER valuation within a sharing community. The aggregator coordinates DER operations in real-time; then it solve this MPEC problem after each billing period. In doing so, the aggregator evaluates two operating costs for each prosumer: the actual cost under coordination and the counter-factual cost if the prosumer independently traded power with the aggregator. We define the difference in these two costs as the coordination surplus, which the aggregator and the prosumer split. Simulation results demonstrate that this sharing procedure effectively achieves the aforementioned goals.

A high-performance nonlinear dynamic scheme for the solution of equilibrium constrained optimization problems

In this paper, a feedback neural network model is proposed to compute the solution of the mathematical programs with equilibrium constraints (MPEC). The MPEC problem is altered into an identical one-level non-smooth optimization problem, then a sequential dynamic scheme that progressively approximates the non-smooth problem is presented. Besides asymptotic stability, it is proven that the limit equilibrium point of the suggested dynamic model is a solution for the original MPEC problem. Numerical simulation of various types of MPEC problems shows the significance of the results. Moreover, the scheme is applied to compute the Stackelberg–Cournot–Nash equilibria.

Saddle point optimality criteria for mathematical programming problems with equilibrium constraints

In this paper, we consider a mathematical programming problem with equilibrium constraints (MPEC). We formulate the Lagrange type dual model for the MPEC and establish weak and strong duality results under convexity assumptions. Further, we investigate the saddle point optimality criteria for the MPEC. We also illustrate our results by an example.

Combating Full-Duplex Active Eavesdropper: A Hierarchical Game Perspective

Security is an issue of paramount importance, yet is it a significant challenge for wireless communications, which becomes more intricate when facing a full duplex (FD) active eavesdropper capable of performing eavesdropping and jamming simultaneously. In this paper, we investigate the physical layer security issue in the presence of an FD active eavesdropper, who launches jamming attacks to further improve the eavesdropping. The jamming, however, also results in self-interference at the eavesdropper itself. This security problem is formulated within a hierarchical game framework where the eavesdropper acts as the leader and the legitimate user is the follower. In particular, we first investigate the follower’s secrecy rate maximization problem and derive the optimal legitimate transmission strategy. Then, the leader’s wiretap rate maximization is expressed as a mathematical program with equilibrium constraints (MPEC). Leveraging the concavity of the follower’s problem, we transform the MPEC problem into a single-level optimization and obtain the jamming power allocation strategy by applying the primal-dual interior-point method. Moreover, we analyze the situations where only partial channel state information is available at the legitimate user and the corresponding impacts on the game. Finally, we present extensive simulation results to validate our theoretical analysis.

Optimal Bidding in Performance-Based Regulation Markets: An MPEC Analysis With System Dynamics

In this paper, we address the problem of optimal bidding in performance-based regulation markets for a large price-maker regulation resource. Focusing on the case of the California Independent System Operator (ISO), detailed market components are considered, such as regulation capacity payment, regulation mileage payment, performance accuracy adjustment, automatic generation control dispatch, and participation factor. Our analysis also incorporates system dynamics of the regulation resource for different resource types and technologies. In principle, our problem formulation is a mathematical program with equilibrium constraints (MPEC). However, our fundamentally new formulations introduce several new challenges in solving the MPEC problem in the context of performance-based regulation markets that are not previously addressed. In fact, global optimization techniques fail to solve the original nonlinear program due to its complexity. Therefore, we undergo several innovative steps to transform the problem into a mixed-integer linear program which is solved with accuracy, reliability, and computational efficiency. Insightful case studies are presented using data from a California ISO regulation market project.

Inverse truss design as a conic mathematical program with equilibrium constraints

We formulate an inverse optimal design problem as a Mathematical Programming problem with Equilibrium Constraints (MPEC). The equilibrium constraints are in the form of a second-order conic optimization problem. Using the so-called Implicit Programming technique, we reformulate the bilevel optimization problem as a single-level nonsmooth nonconvex problem. The major part of the article is devoted to the computation of a subgradient of the resulting composite objective function. The article is concluded by numerical examples demonstrating, for the first time, that the Implicit Programming technique can be efficiently used in the numerical solution of MPECs with conic constraints on the lower level.

Adjustment of the link travel-time functions in traffic equilibrium assignment models

The network calibration problem (NCP) consists of adjusting (or estimating) the parameters of the link travel-time functions in a congested traffic network via the use of traffic link counts and equilibrium costs in a set of origin–destination demand pairs. In this article, a mathematical program with equilibrium constraints (MPEC) is proposed to model this problem. The proposed model is called NCP(ρ) and supposes that the available information has been obtained in several different periods (each period is defined by a single origin–destination demand matrix). The existence of the solutions to the stated NCP(ρ) under weak conditions in the link travel-time functions is established, and a sensitivity analysis of these solutions is carried out. A column generation algorithm (CGA) is proposed to solve the NCP(ρ) model, using sensitivity analysis to prove that this algorithm converges to a local optimum of the MPEC problem. Finally, the special case of estimating the link travel-time functions in additive models when information is only available for a single period is analysed in a numerical study. In this case the NCP(ρ) model is overspecified, that is, it has an infinite number of solutions. For this reason we propose, as an alternative, estimating the congestion costs ψ in a subset of links, rather than estimating the parameterisation of the link travel-time functions ρ. This special case has been named the NCP(ψ) model. The numerical study has been carried out in small and medium-sized networks, with the objective of studying both the proposed MPEC models and the computational aspects of the CGA.

AN IMPLICIT FORMULATION OF MATHEMATICAL PROGRAM WITH COMPLEMENTARITY CONSTRAINTS FOR APPLICATION TO ROBUST STRUCTURAL OPTIMIZATION

This paper discusses an implicit reformulation of a class of MPEC (mathematical program with equilibrium constraints) problems. We particularly focus on an MPEC problem arising from the robust optimization of elastic structures subjected to the uncertain external load. We first review the relation between the worst-case detection and robust constraint satisfaction of the structural responses, and then derive an MPEC formulation of the robust structural optimization. Since a standard constraint qualification is not satisfied at any feasible solution of an MPEC problem, we propose a reformulation based on the smoothed Fischer-Burmeister function, in which the smoothing parameter is treated as an independent variable. Numerical examples of robust structural optimization are presented to demonstrate that the presented formulation can be solved by using a standard nonlinear programming approach.

A neural network approach for solving mathematical programs with equilibrium constraints

A neural network approach is presented for solving mathematical programs with equilibrium constraints (MPEC). The proposed neural network is proved to be Lyapunov stable and capable of generating approximal optimal solution to the MPEC problem. The asymptotic properties of the neural network are analyzed and the condition for asymptotic stability, solution feasibility and solution optimality are derived and the transient behavior of the neural network is simulated and the validity of the network is verified with numerical examples.

New Necessary Optimality Conditions for Bilevel Programs by Combining the MPEC and Value Function Approaches

The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karush-Kuhn-Tucker (KKT) condition and solve the resulting mathematical programming problem with equilibrium constraints (MPEC). In general the classical approach is not valid for nonconvex bilevel programming problems. The value function approach uses the value function of the lower level problem to define an equivalent single level problem. But the resulting problem requires a strong assumption, such as the partial calmness condition, for the KKT condition to hold. In this paper we combine the classical and the value function approaches to derive new necessary optimality conditions under rather weak conditions. The required conditions are even weaker in the case where the classical approach or the value function approach alone is applicable.

MPEC problem formulations and solution strategies with chemical engineering applications

With the development and widespread use of large-scale nonlinear programming (NLP) tools for process optimization, there has been an associated application of NLP formulations with complementarity constraints in order to represent discrete decisions. Also known as mathematical programs with equilibrium constraints (MPECs), these formulations can be used to model certain classes of discrete events and can be more efficient than a mixed integer formulation. However, MPEC formulations and solution strategies are not yet fully developed in process engineering. In this study, we discuss MPEC properties, including concepts of stationarity and linear independence that are essential for well-defined NLP formulations. Nonlinear programming based solution strategies for MPECs are then reviewed and examples of complementarity drawn from chemical engineering applications are presented to illustrate the effectiveness of these formulations.

A new smoothing technique for mathematical programs with equilibrium constraints

A kind of mathematical programs with equilibrium constraints (MPEC) is studied. By using the idea of successive approximation, a smoothing nonlinear programming, which is equivalent to the MPEC problem, is proposed. Thereby, it is ensured that some classical optimization methods can be applied for the MPEC problem. In the end, two algorithm models are proposed with the detail analysis of the global convergence.

A Class of Mathematical Programs with Equilibrium Constraints: A Smooth Algorithm and Applications to Contact Problems

We discuss a special mathematical programming problem with equilibrium constraints (MPEC), that arises in material and shape optimization problems involving the contact of a rod or a plate with a rigid obstacle. This MPEC can be reduced to a nonlinear programming problem with independent variables and some dependent variables implicity defined by the solution of a mixed linear complementarity problem (MLCP). A projected-gradient algorithm including a complementarity method is proposed to solve this optimization problem. Several numerical examples are reported to illustrate the efficiency of this methodology in practice.