Mathematical Program With Complementarity Constraints Research Articles

Formulating and heuristic solving of contact problems in hybrid data-driven computational mechanics

In this work we consider the hybrid Data-Driven Computational Mechanics (DDCM) approach, in which a smooth constitutive manifold is reconstructed to obtain a well-behaved nonlinear optimization problem (NLP) rather than the much harder discrete-continuous NLP (DCNLP) of the direct DDCM approach. The key focus is on the addition of geometric inequality constraints to the hybrid DDCM formulation. Therein, the required constraint force leads to a contact problem in the form of a mathematical program with complementarity constraints (MPCC), a problem class that is still less complex than the DCNLP. For this MPCC we propose a heuristic quick-shot solution approach, which can produce verifiable solutions by solving up to four NLPs. We perform various numerical experiments on three different contact problems of increasing difficulty to demonstrate the potential and limitations of this approach.

Solving Bilevel Programs Based on Lower-Level Mond-Weir Duality

This paper focuses on developing effective algorithms for solving a bilevel program. The most popular approach is to replace the lower-level problem with its Karush-Kuhn-Tucker conditions to generate a mathematical program with complementarity constraints (MPCC). However, MPCC does not satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. In this paper, inspired by a recent work using the lower-level Wolfe duality (WDP), we apply the lower-level Mond-Weir duality to present a new reformulation, called MDP, for bilevel program. It is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense. An example is given to show that, different from MPCC, MDP may satisfy the MFCQ at its feasible points. Relations among MDP, WDP, and MPCC are investigated. On this basis, we extend the MDP reformulation to present another new reformulation (called eMDP), which has similar properties to MDP. Furthermore, to compare two new reformulations with the MPCC and WDP approaches, we design a procedure to generate 150 tested problems randomly and comprehensive numerical experiments show that MDP has quite evident advantages over MPCC and WDP in terms of feasibility to the original bilevel programs, success efficiency, and average CPU time, whereas eMDP is far superior to all other three reformulations. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12071280 and 11901380]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0108 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0108 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

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.

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 relationships are 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.

Lifted stationary points of sparse optimization with complementarity constraints

We aim to compute lifted stationary points of a sparse optimization problem ( $$P_{0}$$ ) with complementarity constraints. We define a continuous relaxation problem ( $$R_{\nu }$$ ) that has the same global minimizers and optimal value with problem ( $$P_{0}$$ ). Problem ( $$R_{\nu }$$ ) is a mathematical program with complementarity constraints (MPCC) and a difference-of-convex objective function. We define MPCC lifted-stationarity of ( $$R_{\nu }$$ ) and show that it is weaker than directional stationarity, but stronger than Clarke stationarity for local optimality. Moreover, we propose an approximation method to solve ( $$R_{\nu }$$ ) and an augmented Lagrangian method to solve its subproblem ( $$R_{\nu ,\sigma }$$ ), which relaxes the equality constraint in ( $$R_{\nu }$$ ) with a tolerance $$\sigma $$ . We prove the convergence of our algorithm to an MPCC lifted-stationary point of problem ( $$R_{\nu }$$ ) and use a sparse optimization problem with vertical linear complementarity constraints to demonstrate the efficiency of our algorithm on finding sparse solutions in practice.

Active fault diagnosis under hybrid bounded and Gaussian uncertainties

In this paper, an active fault diagnosis method is proposed to diagnose faults of linear time-invariant systems under the effect of hybrid bounded and Gaussian uncertainties, which can guarantee an arbitrary confidence level of fault diagnosis. Zonotopes are used to characterize bounded uncertainties. Gaussian uncertainties are handled as ellipsoids by the given confidence coefficient. The confidence domains of output are formulated as the Minkowski sum of a zonotope and an ellipsoid. The fault diagnosis task is done based on the separation of all the confidence domains of different fault situations, which leads to a Mathematical Program with Complementarity Constraints (MPCC). A branch-and-bound method and a smoothing method are provided to solve the program. The effectiveness of the proposed active fault diagnosis method is illustrated by two examples at the end of the paper.

Integration of design and NMPC-based control for chemical processes under uncertainty: An MPCC-based framework

The use of nonlinear model predictive control (NMPC) for the integration of design and control remains as an open area of research. When NMPC is incorporated into the framework for design and control, this results into a bilevel problem formulation. The search of a solution for a bilevel problem is often carried out with the implementation of systematic iterative methodologies. In this work, we implement the classical KKT transformation strategy to transform the original bilevel problem for simultaneous design and NMPC-based control into a single-level dynamic optimization problem. This single-level formulation is referred to as a mathematical program with complementarity constraints (MPCC). Violations to constraint qualifications (CQs) are avoided through the implementation of several reformulation techniques. This facilitates the convergence of the MPCC for optimal design and control. The use of an MPCC-based formulation allows to fully incorporate the NMPC’s necessary conditions for optimality into the design problem as a set of algebraic constraints. Accordingly, the MPCC is solved as a conventional NLP, i.e., we avoid the use of decomposition or simplification methodologies for the solution of the bilevel problem. This guarantees optimality in the solution for integrated design and NMPC-based control. This strategy was implemented in three case studies involving small, medium-scale, and highly nonlinear systems to demonstrate the application of MPCCs for design and NMPC-based control under process disturbances and uncertainty.

M-stationarity for a class of MPCCs in Lebesgue spaces

We show that an optimality condition of M-stationarity type holds for minimizers of a class of mathematical programs with complementarity constraints (MPCCs) in Lebesgue spaces. We apply these results also to local minimizers of an inverse optimal control problem (which is an instance of an infinite-dimensional bilevel optimization problem). The multipliers for the M-stationarity system can be constructed via convex combinations of various multipliers to auxiliary, linear problems. However, proving the existence of the multipliers to these auxiliary problems is difficult and only possible in some situations.

A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints

It is known in the literature that local minimizers of mathematical programs with complementarity constraints (MPCCs) are so-called M-stationary points, if a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds. In this paper we present a new elementary proof for this result. Our proof is significantly simpler than existing proofs and does not rely on deeper technical theory such as calculus rules for limiting normal cones. A crucial ingredient is a proof of a (to the best of our knowledge previously open) conjecture, which was formulated in a Diploma thesis by Schinabeck.

A New Augmented Lagrangian Method for MPCCs—Theoretical and Numerical Comparison with Existing Augmented Lagrangian Methods

We propose a new augmented Lagrangian (AL) method for solving the mathematical program with complementarity constraints (MPCC), where the complementarity constraints are left out of the AL function and treated directly. Two observations motivate us to propose this method: The AL subproblems are closer to the original problem in terms of the constraint structure; and the AL subproblems can be solved efficiently by a nonmonotone projected gradient method, in which we have closed-form solutions at each iteration. The former property helps us show that the proposed method converges globally to an M-stationary (better than C-stationary) point under MPCC relaxed constant positive linear dependence condition. Theoretical comparison with existing AL methods demonstrates that the proposed method is superior in terms of the quality of accumulation points and the strength of assumptions. Numerical comparison, based on problems in MacMPEC, validates the theoretical results.

A new mathematical program with complementarity constraints for optimal localization of pressure reducing valves in water distribution systems

Water loss reduction in water distribution systems (WDSs) is a challenging task for water utilities worldwide. One of the most reliable and cost-effective ways to reduce water loss is to properly regulate operational pressure of the system through optimizing pressure reducing valve (PRV) placements. This well-known engineering problem can be casted into a mixed-integer nonlinear program (MINLP) where binary variables are introduced to represent positions of PRVs. Many works in the literature applied heuristic algorithms to address the optimization problem. In this paper, at first, we proposed a new optimization model and reformulated it as the mathematical program with complementarity constraints (MPCCs). It is due to the fact that the stationary point of the MPCCs is likely to be trapped into bad local solutions, a soft heuristic method is then proposed to determine the MINLP local solution in each iteration before a stationary point of the MPCCs is reached. This method not only enhances the quality of MINLP solution, but also decreases computation time for solving the MPCCs. The newly formulated MPCCs is applied to determine optimal localization of PRVs for two WDS benchmarks and a real-world WDS in Vietnam. The results are compared with others in the literature demonstrating that using our new optimization model, better and more reliable MINLP solution can be found for large scale WDSs.

Local Stackelberg Equilibrium Seeking in Generalized Aggregative Games

We propose a two-layer, semi-decentralized algorithm to compute a local solution to the Stackelberg equilibrium problem in aggregative games with coupling constraints. Specifically, we focus on a single-leader, multiple-follower problem, and after equivalently recasting the Stackelberg game as a mathematical program with complementarity constraints (MPCC), we iteratively convexify a regularized version of the MPCC as inner problem, whose solution generates a sequence of feasible descent directions for the original MPCC. Thus, by pursuing a descent direction at every outer iteration, we establish convergence to a local Stackelberg equilibrium. Finally, the proposed algorithm is tested on a numerical case study involving a hierarchical instance of the charging coordination of Plug-in Electric Vehicles (PEVs).

Robust model predictive control with embedded multi-scenario closed-loop prediction

Robust MPC seeks to mitigate the effects of uncertainty which can lead to suboptimal MPC performance. Previous work on robust MPC includes a stochastic multi-scenario approach to simulate multiple plant realizations and create a control scheme to optimize a performance metric based on the plant scenarios. This paper seeks to combine a scenario-based approach with embedded closed-loop prediction under future MPC control action by directly incorporating MPC subproblems into the overall robust MPC formulation. This allows the current MPC to predict closed-loop MPC responses to a range of uncertain future plant realizations. The resulting multilevel programming problem is solved by reformulating the inner MPC optimization subproblems as algebraic constraints corresponding to their first-order optimality conditions, resulting in a single level mathematical program with complementarity constraints (MPCC). The performance of the robust MPC scheme is evaluated against a standard MPC formulation in linear and nonlinear case studies.

Relations between Abs-Normal NLPs and MPCCs. Part 2: Weak Constraint Qualifications

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a specific slack reformulation suggested in [10], the relations are non-trivial. It turns out that constraint qualifications of Abadie type are preserved. We also prove the weaker result that equivalence of Guginard's (and Abadie's) constraint qualifications for all branch problems hold, while the question of GCQ preservation remains open. Finally, we introduce M-stationarity and B-stationarity concepts for abs-normal NLPs and prove first order optimality conditions corresponding to MPCC counterpart formulations.

Reformulation of the M-Stationarity Conditions as a System of Discontinuous Equations and Its Solution by a Semismooth Newton Method

We show that the Mordukhovich-stationarity system associated with a mathematical program with complementarity constraints (MPCC) can be equivalently written as a system of discontinuous equations w...

Tractable ADMM schemes for computing KKT points and local minimizers for $$\ell _0$$-minimization problems

We consider an $$\ell _0$$ -minimization problem where $$f(x) + \gamma \Vert x\Vert _0$$ is minimized over a polyhedral set and the $$\ell _0$$ -norm regularizer implicitly emphasizes the sparsity of the solution. Such a setting captures a range of problems in image processing and statistical learning. Given the nonconvex and discontinuous nature of this norm, convex regularizers as substitutes are often employed and studied, but less is known about directly solving the $$\ell _0$$ -minimization problem. Inspired by Feng et al. (Pac J Optim 14:273–305, 2018), we consider resolving an equivalent formulation of the $$\ell _0$$ -minimization problem as a mathematical program with complementarity constraints (MPCC) and make the following contributions towards the characterization and computation of its KKT points: (i) First, we show that feasible points of this formulation satisfy the relatively weak Guignard constraint qualification. Furthermore, if f is convex, an equivalence is derived between first-order KKT points and local minimizers of the MPCC formulation. (ii) Next, we apply two alternating direction method of multiplier (ADMM) algorithms, named (ADMM $$_{\mathrm{cf}}^{\mu , \alpha , \rho }$$ ) and (ADMM $$_{\mathrm{cf}}$$ ), to exploit the special structure of the MPCC formulation. Both schemes feature tractable subproblems. Specifically, in spite of the overall nonconvexity, it is shown that the first update can be effectively reduced to a closed-form expression by recognizing a hidden convexity property while the second necessitates solving a tractable convex program. In (ADMM $$_{\mathrm{cf}}^{\mu , \alpha , \rho }$$ ), subsequential convergence to a perturbed KKT point under mild assumptions is proved. Preliminary numerical experiments suggest that the proposed tractable ADMM schemes are more scalable than their standard counterpart while (ADMM $$_{\mathrm{cf}}$$ ) compares well with its competitors in solving the $$\ell _0$$ -minimization problem.

Strongly stable C-stationary points for mathematical programs with complementarity constraints

In this paper we consider the class of mathematical programs with complementarity constraints (MPCC). Under an appropriate constraint qualification of Mangasarian–Fromovitz type we present a topological and an equivalent algebraic characterization of a strongly stable C-stationary point for MPCC. Strong stability refers to the local uniqueness, existence and continuous dependence of a solution for each sufficiently small perturbed problem where perturbations up to second order are allowed. This concept of strong stability was originally introduced by Kojima for standard nonlinear optimization; here, its generalization to MPCC demands a sophisticated technique which takes the disjunctive properties of the solution set of MPCC into account.

Partially Smoothing and Gradient-Based Algorithm for Optimizing the VMI System with Competitive Retailers under Random Demands

Vendor managed inventory (VMI) is an improved sustainable inventory management system, but it is difficult to establish and solve an integrated Stackelberg game model under the complicated practical environment. In this paper, a bilevel programming model is proposed to formulate the VMI system by taking into account the uncertainty of demand, the competition among retailers, the cooperative advertising, the shortage and holding costs, and the practical constraints. For the established stochastic model being associated with continuously random demands, a deterministic mathematical program with complementarity constraints (MPCC) is first derived by expectation method and the first-order optimality conditions of the lower-level problem. Then, with a partially smoothing technique, the MPCC is solved by transforming it into a series of standard smooth optimization subproblems. Finally, owing to complexity caused by evaluating the integrals with unknown decision variables in the objective function, an efficient algorithm is developed to solve the problem based on the gradient information of model. Sensitivity analysis has been employed to reveal a number of managerial implications from the constructed model and algorithm. (1) The participation rate depends on advertising expenditures from both the manufacturer and the retailer. There exists an optimal threshold of participation rate for the manufacturer, which can be provided by the intersection point of the manufacturer and retailer’s cost-profit curves. (2) The manufacturer’s advertising policy is less sensitive to uncertainty of demand than the change of the retailer’s advertising policy. (3) The manufacturer in the VMI system should concern about the differences caused by symmetric or asymmetric retailers.

The ℓ1—Exact Penalty-Barrier Phase for Degenerate Nonlinear Programming Problems in Ipopt

Failure to satisfy Constraint Qualifications (CQs) leads to serious convergence difficulties for state-of-the-art Nonlinear Programming (NLP) solvers. Since this failure is often overlooked by practitioners, a strategy to enhance the robustness properties for problems without CQs is vital. Inspired by penalty merit functions and barrier-like strategies, we propose and implement a combination of both in Ipopt. This strategy has the advantage of consistently satisfying the Linear Independence Constraint Qualification (LICQ) for an augmented problem, readily enabling regular step computations within the interior-point framework. Additionally, an update rule inspired by the work of Byrd et al. (2012) is implemented, which provides a dynamic increase of the penalty parameter as stationary points are approached. Extensive test results show favorable performance and robustness increases for our ℓ1 —penalty strategies, when compared to the regular version of Ipopt. Moreover, a dynamic optimization problem with nonsmooth dynamics formulated as a Mathematical Program with Complementarity Constraints (MPCC) was solved in a single optimization stage without additional reformulation. Thus, this ℓ1— strategy has proved useful for a broad class of degenerate NLPs.

Proactive Energy Optimization in Residential Buildings with Weather and Market Forecasts

This work explores the development of a home energy management system (HEMS) that uses weather and market forecasts to optimize the usage of home appliances and to manage battery usage and solar power production. A Moving Horizon Estimation (MHE) application is used to find the unknown home model parameters. These parameters are then updated in a Model Predictive Controller (MPC) which optimizes and balances competing comfort and economic objectives. Combining MHE and MPC applications alleviates model complexity commonly seen in HEMS by using a lumped parameter model that is adapted to fit a high-fidelity model. Heating, ventilation, and air conditioning (HVAC) on/off behaviors are simulated by using Mathematical Program with Complementarity Constraints (MPCCs) and solved in near real time with a non-linear solver. Removing HVAC on/off as a discrete variable and replacing it with an MPCC reduces solve time. The results of this work indicate that energy management optimization significantly decreases energy costs and balances energy usage more effectively throughout the day. A case study for Phoenix, Arizona shows an energy reduction of 21% and a cost reduction of 40%. This simulated home contributes less to the grid peak load and therefore improves grid stability and reduces the amplitude of load-following cycles for utilities. The case study combines renewable energy, energy storage, forecasts, cooling system, variable rate electricity plan and a multi-objective function allowing for a complete home energy optimization assessment. There remain several challenges, including improved forecast models, improved computational performance to allow the algorithms to run in real time, and mixed empirical/physics-based machine-learning methods to guide the model structure.