Complementarity Constraints Research Articles

Discrete optimal power flow with prohibited zones, multiple-fuel options, and practical operational rules for control devices

Although the optimal power flow (OPF) problem has been extensively studied, solving realistic OPF models that accurately represent the operating behavior of power system components remains challenging. This paper proposes a novel model for the AC OPF problem, aiming to minimize the fuel costs of thermal units while taking into account valve-point loading effects (VPLE), prohibited operation zones (POZ), multiple fuel options (MFO), and operational rules associated with the discrete tap ratios of on-load tap changer (OLTC) transformers and with the discrete shunt susceptances of capacitor/reactor banks. These rules are represented using complementarity constraints. We propose a solution approach that integrates several strategies to address the non-smooth features of the objective function related to VPLE, the disjoint constraints and functions tied to POZ and MFO, the discrete characteristics of the reactive control variables, and the complementarity constraints governing operational rules linked to voltage control devices such as OLTC transformers and capacitor/reactor banks. The resulting optimization problem is designed to be compatible with commercial solver packages. Numerical tests on the IEEE 30, 118, and 300-bus systems aim to examine the cumulative impact of these operational factors on the optimal solution. The solution strategy proposed has demonstrated its effectiveness in solving the proposed OPF problem within reasonable computation times.

An AC-Feasible Linear Model in Distribution Networks With Energy Storage

With the increasing deployment of distributed energy resources (DERs), dispatching DERs subject to operational constraints in distribution networks draws much attention. One challenge is the non-convexities in 1) system-wide AC power flow constraints and 2) the individual complementarity constraint of energy storage. To resolve this challenge, this paper studies an AC-feasible linear model in distribution networks with energy storage, including its formulation, analysis, and applications. First, an AC-feasible linear model is formulated as a set of linear constraints on controllable DERs and uncontrollable power demand by 1) converting the non-convex system-wide constraints into linear constraints based on the Brouwer's fixed-point theorem and the second-order Taylor expansion, and 2) replacing the non-convex individual complementarity constraint of energy storage with one properly designed linear constraint. Furthermore, to analyze the power demand level at which the proposed linear model can provide a solution, this paper proposes an examination-based projection method under the Monte Carlo framework to handle projections of thousands of dimensions from linear constraints over time periods. Finally, the potential applications of our AC-feasible linear model are discussed. Numerical experiments are conducted in the IEEE 33-bus and 136-bus test systems to demonstrate the effectiveness of the proposed methods.

Time-Optimal Handover Trajectory Planning for Aerial Manipulators Based on Discrete Mechanics and Complementarity Constraints

Time-Optimal Handover Trajectory Planning for Aerial Manipulators Based on Discrete Mechanics and Complementarity Constraints

Distributionally Robust Model Predictive Control for Smart Electric Vehicle Charging Station With V2G/V2V Capability

This paper proposes a distributionally robust model predictive control (DRMPC) for energy management of a vehicle-to-grid (V2G)/vehicle-to-vehicle (V2V)-enabled smart electric vehicle charging station (EVCS) with a photovoltaic (PV) system and an energy storage system. The proposed DRMPC method aims to reduce the operational cost of the EVCS while ensuring the desired charging demands of electric vehicle (EV) users under uncertainties in electricity buying/selling prices, PV generation outputs, and future EV charging demands. To cope with these uncertainties, the proposed method includes the following three features: i) tractable reformulation of the worst-case expected buying cost and selling revenue using a Wasserstein metric and duality theory, ii) determination of a distributionally robust bound on the random PV generation output using its support information, and iii) a scenario-based approach to predicting the future EV charging demand. To improve computational efficiency, a penalty method is proposed to relax the complementarity constraints, while still ensuring nonsimultaneous charging and discharging of EVs under the derived sufficient conditions. Numerical examples using a real-world operational dataset of the EVCS are provided to demonstrate the effectiveness of the proposed DRMPC method under uncertain environments in terms of the EVCS cost saving via V2G/V2V capability, data utilization, and computational complexity.

BioSTD: A New Tensor Multi-View Framework via Combining Tensor Decomposition and Strong Complementarity Constraint for Analyzing Cancer Omics Data.

Advances in omics technology have enriched the understanding of the biological mechanisms of diseases, which has provided a new approach for cancer research. Multi-omics data contain different levels of cancer information, and comprehensive analysis of them has attracted wide attention. However, limited by the dimensionality of matrix models, traditional methods cannot fully use the key high-dimensional global structure of multi-omics data. Moreover, besides global information, local features within each omics are also critical. It is necessary to consider the potential local information together with the high-dimensional global information, ensuring that the shared and complementary features of the omics data are comprehensively observed. In view of the above, this article proposes a new tensor integrative framework called the strong complementarity tensor decomposition model (BioSTD) for cancer multi-omics data. It is used to identify cancer subtype specific genes and cluster subtype samples. Different from the matrix framework, BioSTD utilizes multi-view tensors to coordinate each omics to maximize high-dimensional spatial relationships, which jointly considers the different characteristics of different omics data. Meanwhile, we propose the concept of strong complementarity constraint applicable to omics data and introduce it into BioSTD. Strong complementarity is used to explore the potential local information, which can enhance the separability of different subtypes, allowing consistency and complementarity in the omics data to be fully represented. Experimental results on real cancer datasets show that our model outperforms other advanced models, which confirms its validity.

New Constraint Qualifications for Mathematical Programs with Second-Order Cone Complementarity Constraints

New Constraint Qualifications for Mathematical Programs with Second-Order Cone Complementarity Constraints

Bilevel Imaging Learning Problems as Mathematical Programs with Complementarity Constraints: Reformulation and Theory

Bilevel Imaging Learning Problems as Mathematical Programs with Complementarity Constraints: Reformulation and Theory

HL-nets: Physics-informed neural networks for hydrodynamic lubrication with cavitation

Recently, physics-informed neural networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs). In this study, we establish a deep learning computational framework, HL-nets, for computing the flow field of hydrodynamic lubrication involving cavitation effects. The Swift-Stieber (SS) and the Jakobsson-Floberg-Olsson (JFO) cavitation conditions are implemented in the PINNs to solve the Reynolds equation. For the non-negativity constraint of the SS cavitation condition, a penalizing scheme with a residual of the non-negativity and an imposing scheme with a continuous differentiable non-negative function are proposed. For the complementarity constraint of the JFO cavitation condition, the pressure and cavitation fraction are taken as the neural network outputs, and the residual of the Fischer-Burmeister (FB) equation constrains their complementary relationships. Multi-task learning (MTL) methods are applied to balance the loss terms of functions and constraints described above. To estimate the accuracy of HL-nets, we present a numerical solution of the Reynolds equation for oil-lubricated bearings involving cavitation. The results indicate that the proposed HL-nets can highly accurately simulate hydrodynamic lubrication involving cavitation phenomena. The imposing scheme can effectively improve the accuracy of the training results of PINNs, and it is expected to have great potential to be applied to different fields where the non-negativity constraint is needed.

Integrated deployment of dedicated lane and roadside unit considering uncertain road capacity under the mixed-autonomy traffic environment

The rapid development of automated vehicle (AV) technologies enables us to explore how to update the urban road infrastructure to cater for the upcoming mixed-autonomy traffic with both AVs and human-driven vehicles (HVs). In this study, we aim to seek an optimal solution to integrate the deployment of AV-dedicated lane and roadside unit assisting automated driving subject to a limited budget by considering the route choice behaviors of AVs and HVs. An AV-dedicated lane segregates AVs from the mixed traffic to create a fully automated driving environment and eliminate disturbances from HVs. The deployment of roadside units assisting automated driving could overcome the connectivity gap for AVs and lower their headway when they follow an HV. We seek to develop a robust and optimal integrated deployment solution that accommodates the uncertain road capacity caused by the stochastic mixed AV and HV fleet sequence. We first establish the stochastic network equilibrium conditions for the mixed traffic, and formulate the integrated deployment problem as a mathematical program with complementarity constraints (MPCC). The MPCC is approximated by a mixed-integer linear programing (MILP) model, which allows existing algorithms for its global optimum. We further develop two interesting strategies, including domain reduction and breakpoint selection, to enhance the effectiveness of the MILP model. Numerical experiments are finally carried out to evaluate the feasibility of research methodology proposed in this study and find some valuable insights.

A Global Optimization Approach for Multimarginal Optimal Transport Problems with Coulomb Cost

In this work, we construct a novel numerical method for solving the multi-marginal optimal transport problems with Coulomb cost. This type of optimal transport problems arises in quantum physics and plays an important role in understanding the strongly correlated quantum systems. With a Monge-like ansatz, the orginal high-dimensional problems are transferred into mathematical programmings with generalized complementarity constraints, and thus the curse of dimensionality is surmounted. However, the latter ones are themselves hard to deal with from both theoretical and practical perspective. Moreover in the presence of nonconvexity, brute-force searching for global solutions becomes prohibitive as the problem size grows large. To this end, we propose a global optimization approach for solving the nonconvex optimization problems, by exploiting an efficient proximal block coordinate descent local solver and an initialization subroutine based on hierarchical grid refinements. We provide numerical simulations on some typical physical systems to show the efficiency of our approach. The results match well with both theoretical predictions and physical intuitions, and give the first visualization of optimal transport maps for some two dimensional systems.

Quadratic regularization of bilevel pricing problems and application to electricity retail markets

We consider the profit-maximization problem solved by an electricity retailer who aims at designing a menu of contracts. This is an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A basic approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second classical approach is to consider logit model to add a probabilistic behavior in the customers’ choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer’s response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer’s response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop a pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming, showing the effectiveness of the approach. Throughout the paper, the electricity retailer problem is our guideline, and we present a numerical study on this application case.

On solving difference of convex functions programs with linear complementarity constraints

On solving difference of convex functions programs with linear complementarity constraints

Guaranteeing a Physically Realizable Battery Dispatch Without Charge-Discharge Complementarity Constraints

The non-convex complementarity constraints present a fundamental computational challenge in energy constrained optimization problems. In this work, we present a new, linear, and robust battery optimization formulation that sidesteps the need for battery complementarity constraints and integers and prove analytically that the formulation guarantees that all energy constraints are satisfied which ensures that the optimized battery dispatch is physically realizable. In addition, we bound the worst-case model mismatch and discuss conservativeness. Simulation results further illustrate the effectiveness of this approach.

The exactness of the [formula omitted] penalty function for a class of mathematical programs with generalized complementarity constraints

In a Mathematical Program with Generalized Complementarity Constraints (MPGCC), complementarity relation is imposed between each pair of variable blocks. MPGCC includes the traditional Mathematical Program with Complementarity Constraints (MPCC) as a special case. On account of the disjunctive feasible region, MPCC and MPGCC are generally difficult to handle. The ℓ1 penalty method, often adopted in computation, opens a way of circumventing the difficulty. Yet it remains unclear about the exactness of the ℓ1 penalty function, namely, whether there exists a sufficiently large penalty parameter so that the penalty problem shares the optimal solution set with the original one. In this paper, we consider a class of MPGCCs that are of multi-affine objective functions. This problem class finds applications in various fields, e.g., the multi-marginal optimal transport problems in many-body quantum physics and the pricing problems in network transportation. We first provide an instance from this class, the exactness of whose ℓ1 penalty function cannot be derived by existing tools. We then establish the exactness results under rather mild conditions. Our results cover those existing ones for MPCC and apply to multi-block contexts.

A Decomposition Algorithm with Fast Identification of Critical Contingencies for Large-Scale Security-Constrained AC-OPF

The article “A Decomposition Algorithm with Fast Identification of Critical Contingencies for Large-Scale Security-Constrained AC-OPF” presents the decomposition algorithm used by Team GO-SNIP for the ARPA-E Grid Optimization (GO) Competition Challenge 1, held from November 2018 through October 2019. The algorithm involves unique contingency ranking and evaluation strategies for determining the important contingencies to include in a master problem that approximates the original large-scale security-constrained problem. It also involves efficient strategies for handling the complementarity constraints that appear in the model and for handling the arising degeneracies. Software implementation details are described, and the results of an extensive set of numerical experiments are provided to illustrate the effectiveness of each of the used techniques. Team GO-SNIP received a second-place finish in Challenge 1 of the Go Competition.

Joint location and pricing optimization of self-service in urban logistics considering customers’ choice behavior

Self-services by widely-deployed stations allow customers to pick up their packages themselves and have great potential to reduce last-mile delivery cost by aggregating spatial demands. An efficient self-service design for the locations of self-service stations and the service price will help to shift customers from the home-delivery service to the cost-effective self-service, thus reducing the vehicle routing cost of the dedicated fleet. This study proposes a comprehensive modeling and optimization framework for the joint self-service station location and service price (SLSP) optimization problem considering customers’ choice behavior and vehicle routing in order to maximize the profit of the logistics service providers in the last-mile delivery system. Logit equilibrium and Wardrop equilibrium conditions are employed to characterize the delivery service choice and route choice behavior of customers. A convex optimization model is developed to capture the distribution of customers opting for home-delivery service and self-services at various locations, who aim to maximize their expected utilities. We then formulate the SLSP problem as a mathematical program with complementarity constraints. An active set algorithm is proposed to solve the model effectively. Numerical experiments based on the Nguyen-Dupuis and Sioux Falls networks are conducted to demonstrate the proposed framework and derive managerial insights.

Multi-area transmission maintenance scheduling for two-level markets

With the increasing renewable energy and expanding trans-regional transmission, higher requirements are placed on the outage maintenance scheduling for high-voltage power transmission networks. In this paper, a bi-level multi-area joint maintenance scheduling optimization model is established for high-voltage transmission lines taking into account the consumption of renewable energy under two-level market environment. This model is able to achieve the following in terms of the maintenance scheduling: (i) It adapts to the optimization objectives and modes of different electricity spot markets between and within areas; (ii) The goals of maintenance willingness and consumption of renewable energy are balanced; (iii) By using the Kuhn–Tucker method, the bi-level model is transformed into a single-level mixed integer programming problem with complementarity constraints for the purpose of solving. The method proposed in this paper is verified by numerical examples, which reveal how the proposed model can improve the capability of multi-area power system to accommodate renewable energy and better arrange the maintenance resources.

Pathfollowing for parametric mathematical programs with complementarity constraints

In this paper we study procedures for pathfollowing parametric mathematical programs with complementarity constraints. We present two algorithms, one based on the penalty approach for solving standalone MPCCs, and one based on tracing active set bifurcations arising from doubly-active complementarity constraints. We demonstrate the performance of these two approaches on a variety of examples with different types of stationary points and also a simple engineering problem with phase changes.

The Augmented Lagrangian Method for Mathematical Programs with Vertical Complementarity Constraints Based on Inexact Scholtes Regularization

Mathematical program with vertical complementarity constraints (MPVCC) plays an important role in economics and engineering. Due to the vertical complementarity structure, most of the standard constraint qualifications fail to hold at a feasible point. Without constraint qualifications, the Karush–Kuhn–Tucker (KKT) conditions are not necessary optimality conditions. The classical methods for solving constrained optimization problems applied to MPVCC are likely to fail. It is necessary to establish efficient algorithms for solving MPVCC from both theoretical and numerical points of view. We present an algorithm to obtain stationarity of MPVCC by solving a sequence of Scholtes regularized problems. We consider the Scholtes regularization method with the sequence of approximate KKT points only. We prove that, under strictly weaker constraint qualifications, the accumulation point of the approximate KKT points is Clarke (C-) stationary point. In particular, we can get Mordukhovich (M-) or strongly (S-) stationary point under additional assumptions. From these results, we apply an augmented Lagrangian method to obtain a solution of MPVCC and give the convergence analysis. In particular, the accumulation point of the generated iterates is an S-stationary point if some boundedness conditions hold. The numerical results show that it is an effective way to solve MPVCC.

Unified Terrestrial/Aerial Motion Planning for HyTAQs via NMPC

In this study, an NMPC-based planning approach is proposed for Hybrid Terrestrial and Aerial Quadrotors (HyTAQs) to achieve safe and efficient hybrid locomotion in unknown environments. The approach ensures bimodal dynamic feasibility and fully exploits the high efficiency and accessibility of HyTAQs. Equipped with the complementarity constraints for hybrid dynamics, the unified NMPC enables simultaneous optimization of the full-state trajectory and the inherent locomotion mode without the computational burden of mixed-integer programming. Also, the uncertainty bounds are evaluated along the nominal trajectory in a receding horizon manner and incorporated within the geometry constraints for the safety and robustness. Further, the bimodal energy efficiency is fully considered by the topology-guided path searching, achieving more efficient mode selection. Then, a compact unified terrestrial-aerial planner is implemented on the HyTAQ and validated through various comparisons and navigation tests. Experimental results verify the high efficiency and robustness of the proposed approach.