Nonconvex Mathematical Programming Research Articles

Solving a bilevel strategic offering model for a generation company in a hydrothermal energy market: A new convexification approach

This paper proposes a bilevel strategic offering (SO) model suitable for a strategic company (SC) that operates in a hydro-dominated day-ahead energy market. The upper-level model of the proposed SO model maximizes the SC’s profits, while the lower-level model embodies a hydrothermal market clearing (MC). This MC incorporates detailed hydro and thermal constraints for units of both the SC and non-strategic companies (NSC), in addition to the network constraints. The proposed solution approach entails replacing the lower-level model with its corresponding primal and dual constraints, along with a strong duality equality constraint. This leads to a non-convex mathematical programming with equilibrium constraints (MPEC) model, which is then reformulated to feature only a single bilinear term in its structure. Finally, the recast MPEC model is convexified around an initial point and solved iteratively until a convergence criteria is achieved. The model and solution approach proposed are evaluated on an adapted version of the IEEE 24-bus system, exploring scenarios where the SC acts strategically or non-strategically (perfect competition).

Partial reformulation-linearization based optimization models for the Golomb ruler problem

In this paper, we provide a straightforward proof of a conjecture proposed in \cite{Duxbury2021} regarding the optimal solutions of a non-convex mathematical programming model of the Golomb ruler problem. Subsequently, we investigate the computational efficiency of four new binary mixed-integer linear programming models to compute optimal Golomb rulers. These models are derived from a well-known nonlinear integer model proposed in \cite{Kocuk2019}, utilizing the reformulation-linearization technique. \modif{Finally, we provide the correct outputs of the greedy heuristic proposed in \cite{Duxbury2021} and correct some false conclusions stated or implied therein.}

Optimality conditions for nonconvex mathematical programming problems using weak subdifferentials and augmented normal cones

Optimality conditions for nonconvex mathematical programming problems using weak subdifferentials and augmented normal cones

An exact optimization method for coordinating the arrival times of urban rail lines at a common corridor

The trips of a high-frequency urban rail line are typically planned with the aim of achieving even time headways. This results in reliable services for each urban rail line, where successive trips have the same time headway. Maintaining even time headways for each service line has significant advantages for the passengers of the line, but it might result in safety issues, vehicle bunching, and increased transfer times at a common corridor served by multiple urban rail lines. This study investigates the problem of urban rail corridor coordination and develops an exact optimization method for coordinating the vehicle trips of different lines that serve stations along a joint corridor. The proposed formulation is a non-convex mathematical program which is reformulated as a mixed-integer quadratic program with a convex objective function. A branch-and-bound algorithm coupled with the Active-set method is proposed for solving the model to global optimality. Results from a toy network and a case study of the light rail service in The Hague, The Netherlands, demonstrate the improvement potential of time headways at a common corridor, while accounting for the effect on the variation of time headways at isolated segments of the individual service lines.

DQN based user association control in hierarchical mobile edge computing systems for mobile IoT services

This paper proposes a new user association control method to maximize system utility in terms of system throughput, edge throughput, and handover cost in mmWave-based HMEC (mm-HMEC) systems. We first formulated a non-convex mathematical programming model and then proposed centralized and distributed deep Q-network (DQN)-based range expansion bias control algorithms. We discovered that the proposed schemes provided polynomial communication overhead and computation complexity. Through simulations, we evaluated the proposed centralized and distributed schemes under various user equipment (UE) mobility models: random waypoint mobility model (RWM), random direction mobility model (RDM), and Manhattan mobility model (MAN). We first confirmed that the proposed distributed control achieves almost the same performance as the proposed centralized control. Second, with 10 slave-MEC servers, the proposed distributed control provides 212.17%, 62.07%, 46.36%, and 48.63% enhanced utility; 33.68%, 18.19%, 9.28%, and 12.66% enhanced average cell throughput; 14.32%, 38.32%, 63.87%, and 24.57% enhanced edge throughput; and 39.09%, 25.42%, 29.32%, and 25.95% reduced handover cost compared to random cell range expansion (CRE), standard CRE, dynamic CRE, and Q-learning-based CRE, respectively.

A study on vector variational-like inequalities using convexificators and application to its bi-level form

<p style='text-indent:20px;'>This paper deals with the weak versions of the vector variational-like inequalities, namely Stampacchia and Minty type under invexity in the framework of convexificators. The connection between both the problems along with the link to vector optimization problem are analyzed. An application to nonconvex mathematical programming has also been presented. Further, the bi-level version of these problems is formulated and a procedure to obtain the solution involving the auxiliary principle technique is described in detail. We have shown that the iterative algorithm with the help of which we get the approximate solution converges strongly to the exact solution of the problem.</p>

Optimality conditions for mathematical programs with equilibrium constraints using directional convexificators

We consider a nonsmooth and nonconvex mathematical program with equilibrium constraints (MPEC) where its functions are not necessarily locally Lipschitz/continuous. In exploiting the idea of directional convexificators introduced by Dempe and Pelicka [Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming. J Global Optim. 2015;61:769–788], we define two forms of nonsmooth versions of the classical Abadie constraint qualifications, and introduce nonsmooth stationarity conditions, which are based on directional convexificators. Then, we employ these conditions to derive first-order optimality conditions subject to generalized Abadie constraint qualifications. Finally, we establish sufficient optimality conditions for MPEC under a new assumption of generalized convexity.

Robust duality for generalized convex nonsmooth vector programs with uncertain data in constraints

Robust optimization has come out to be a potent approach to study mathematical problems with data uncertainty. We use robust optimization to study a nonsmooth nonconvex mathematical program over cones with data uncertainty containing generalized convex functions. We study sufficient optimality conditions for the problem. Then we construct its robust dual problem and provide appropriate duality theorems which show the relation between uncertainty problems and their corresponding robust dual problems.

Global optimization for bilevel portfolio design: Economic insights from the Dow Jones index

This paper deals with a portfolio selection problem with transaction costs and two levels of decision-making. It is assumed that the decision making structure is twofold: there is a broker-dealer that controls the fees to be charged on the different securities in order to maximize his benefit and there is an investor who chooses his portfolio trying to minimize risk while ensuring a minimum level of return. This structure gives rise to an implicit hierarchical competition that consists in anticipating the rational decision of the other agent in order to optimize the decision-makers’ own criteria. We analyze different situations depending on who is first in the hierarchy: the broker-dealer or the investor. We present different nonlinear and nonconvex mathematical programming models for the different situations and develop an extensive computational study in which we discuss the ensuing economic insights for the models based on Dow Jones index data.

Characterization of generalized FJ and KKT conditions in nonsmooth nonconvex optimization

In this paper, we investigate new generalizations of Fritz John (FJ) and Karush–Kuhn–Tucker (KKT) optimality conditions for nonconvex nonsmooth mathematical programming problems with inequality constraints and a geometric constraint set. After defining generalized FJ and KKT conditions, we provide some alternative-type characterizations for them. We present characterizations of KKT optimality conditions without assuming traditional Constraint Qualification (CQ), invoking strong duality for a sublinear approximation of the problem in question. Such characterizations will be helpful when traditional CQs fail. We present the results with more details for a problem with a single-inequality constraint, and address an application of the derived results in mathematical programming problems with equilibrium constraints. The objective function and constraint functions of the dealt with problem are nonsmooth and we establish our results in terms of the Clarke generalized directional derivatives and generalized gradient. The results of the current paper cover classic optimality conditions existing in the literature and extend the outcomes of Flores-Bazan and Mastroeni (SIAM J Optim 25:647–676, 2015).

Augmented Lagrangian Approach to the Newsvendor Model with Component Commonality

Component commonality is a well-known approach in manufacturing, where the same components are used for multiple products. It has been implemented by many established companies such as Airbus, Kodak, Toyota, etc. We consider a standard two-product inventory model with a common component. The demands for the products are independent random variables. Instead of the usual approach to minimize the total shortage quantity, we propose to minimize the total shortage cost. The resulting problem is a non-convex nonlinear mathematical program. We illustrate the use of a primal-dual proximal method to solve this problem by obtaining numerically the optimal allocations of components. In particular, we show that a higher unit shortage cost induces a higher allocation.

Bilevel programming methods for computing single-leader-multi-follower equilibria in normal-form and polymatrix games

The concept of leader-follower (or Stackelberg) equilibrium plays a central role in a number of real-world applications bordering on mathematical optimization and game theory. While the single-follower case has been investigated since the inception of bilevel programming with the seminal work of von Stackelberg, results for the case with multiple followers are only sporadic and not many computationally affordable methods are available. In this work, we consider Stackelberg games with two or more followers who play a (pure or mixed) Nash equilibrium once the leader has committed to a (pure or mixed) strategy, focusing on normal-form and polymatrix games. As customary in bilevel programming, we address the two extreme cases where, if the leader’s commitment originates more Nash equilibria in the followers’ game, one which either maximizes (optimistic case) or minimizes (pessimistic case) the leader’s utility is selected. First, we show that, in both cases and when assuming mixed strategies, the optimization problem associated with the search problem of finding a Stackelberg equilibrium is mathcal {NP}-hard and not in Poly-mathcal {APX} unless mathcal {P} = mathcal {NP}. We then consider different situations based on whether the leader or the followers can play mixed strategies or are restricted to pure strategies only, proposing exact nonconvex mathematical programming formulations for the optimistic case for normal-form and polymatrix games. For the pessimistic problem, which cannot be tackled with a (single-level) mathematical programming formulation, we propose a heuristic black-box algorithm. All the methods and formulations that we propose are thoroughly evaluated computationally.

Short-Term Forecasting of Price-Responsive Loads Using Inverse Optimization

We consider the problem of forecasting the aggregate demand of a pool of price-responsive consumers of electricity. The response of the aggregate load to price is modeled by an optimization problem that is characterized by a set of marginal utility curves and minimum and maximum power consumption limits. The task of estimating these parameters is addressed using a generalized inverse optimization scheme that, in turn, requires solving a nonconvex mathematical program. We introduce a solution method that overcomes the nonconvexities by solving instead two linear problems with a penalty term, which is statistically adjusted by using a cross-validation algorithm. The proposed methodology is data-driven and leverages information from regressors, such as time and weather variables, to account for changes in the parameter estimates. The power load of a group of heating, ventilation, and air conditioning systems in buildings is simulated, and the results show that the aggregate demand of the group can be successfully captured by the proposed model, making it suitable for short-term forecasting purposes.

Randomized Decomposition Solver with the Quadratic Assignment Problem as a Case Study

This paper presents a new local search approach, called randomized decomposition (RD), for solving nonlinear, nonconvex mathematical programs. Starting from a feasible solution, RD partitions the problem’s decision variables into a randomly ordered list of randomly generated subsets. RD then optimizes over the variables in each subset, keeping all other variables fixed. Unlike most other decomposition methods, no knowledge of the problem structure is required. RD has been combined with a metaheuristic RDPerturb, for escaping local optima, to create a generic framework for solving mathematical programs, especially hard combinatorial nonconvex problems. The framework has been implemented as an optimization platform we call RDSolver and successfully applied to over 400 instances of the quadratic assignment problem (QAP). The results obtained by RDSolver are competitive with the solutions obtained by heuristics specially tailored for those problems, even though RDSolver is a general purpose mathematical progr...

FISCAL PERFORMANCE BENCHMARKING OF INDIAN STATES-A ROBUST FRONTIER APPROACH

Aim: The objective of the paper is to construct an index of fiscal performance of Indian states using four non-parametric approaches ( Data Envelopment Analysis(DEA), Free Disposal Hull(FDH), Order-m and Order-alpha).The reason behind using non-parametric methods for the purpose of construction of index is that the traditional ratio approach is incapable of handling multiple input and output indicators.Research methods: The present study uses a two stage approach. In the first stage, four non-parametric methods have been used to evaluate the performance of Indian states for five consecutive years. The input and output indicators have been selected on the basis of a simple theoretical model. Further, in order to tackle the problem of estimation bias (due to sampling variations) bootstrapped DEA and bootstrapped Order-m methods have been applied. In the second stage ,impact of indebtedness on the performance of the states has been assessed using a censored regression framework.Findings: The major outcome of the study is the construction of a fiscal performance index based on multiple indicators. Moreover, the second stage results indicate that state performance is significantly influenced by their degree of indebtedness.Originality/Value of the article: The present study is perhaps the first attempt to assess the performance of sub-national units in terms of both convex and non-convex mathematical programming methods.Implications of the research : The approach (with suitable modifications) can be effectively used to benchmark state performance which can serve as a basis for resource transfer from the central government to the states.

On Condition Number Theorems in Mathematical Programming

A condition number of nonconvex mathematical programming problems is defined as a measure of the sensitivity of their global optimal solutions under canonical perturbations. A (pseudo-)distance among problems is defined via the corresponding augmented Kojima functions. A characterization of well-conditioning is obtained. In the nonconvex case, we prove that the distance from ill-conditioning is bounded from above by a multiple of the reciprocal of the condition number. Moreover, a lower bound of the distance from a special class of ill-conditioned problems is obtained in terms of the condition number. The proof is based on a new theorem about the permanence of the Lipschitz character of set-valued inverse mappings. A uniform version of the condition number theorem is proved for classes of convex problems defined through bounds of some constants available from problem’s data.

Optimization of incentive polices for plug-in electric vehicles

High purchase prices and the lack of supporting infrastructure are major hurdles to the adoption of plug-in electric vehicles (PEVs). It is widely recognized that the government could help break these barriers through incentive policies, such as offering rebates to PEV buyers or funding charging stations. The objective of this paper is to propose a modeling framework that can optimize the design of such incentive policies. The proposed model characterizes the impact of the incentives on the dynamic evolution of PEV market penetration over a discrete set of time intervals, by integrating a simplified consumer vehicle choice model and a macroscopic travel and charging model. The optimization problem is formulated as a nonlinear and non-convex mathematical program and solved by a specialized steepest descent direction algorithm. We show that, under mild regularity conditions, the KKT conditions of the proposed model are necessary for local optimum. Results of numerical experiments indicate that the proposed algorithm is able to obtain satisfactory local optimal policies quickly. These optimal policies consistently outperform the alternative policies that mimic the state-of-the-practice by a large margin, in terms of both the total savings in social costs and the market share of PEVs. Importantly, the optimal policy always sets the investment priority on building charging stations. In contrast, providing purchase rebates, which is widely used in current practice, is found to be less effective.

A Characterization of Stationary Nash Equilibria of Single Controller Constrained Stochastic Games

We consider a two player finite state-action general sum single controller constrained stochastic game with both discounted and average cost criteria. We consider the situation where player 1 has subscription-based constraints and player 2, who controls the transition probabilities, has realization-based constraints which can also depend on the strategies of player 1. It is known that a stationary Nash equilibrium for discounted case exists under strong Slater condition, while, for the average case, stationary Nash equilibrium exists if additionally the Markov chain is unichain. For each case we show that the set of stationary Nash equilibria of this game has one to one correspondence with the set of global minimizers of a certain nonconvex mathematical program. If the constraints of player 2 do not depend on the strategies of player 1, then the mathematical program reduces to a quadratic program. The known linear programs for zero sum games of this class can be obtained as a special case of above quadratic programs.

Characterizing FJ and KKT Conditions in Nonconvex Mathematical Programming with Applications

In this paper we analyze the Fritz John and Karush--Kuhn--Tucker (KKT) conditions for a (Gâteaux) differentiable nonconvex optimization problem with inequality constraints and a geometric constraint set. The Fritz John condition is characterized in terms of an alternative theorem which covers beyond standard situations, while characterizations of KKT conditions, without assuming constraints qualifications, are related to strong duality of a suitable linear approximation of the given problem and the properties of its associated image mapping. Such characterizations are suitable for dealing with some problems in structural optimization, where most of the known constraint qualifications fail. In particular, several examples are given showing the usefulness and optimality, in a certain sense, of our results, which provide much more information than those (including the Mordukhovich normal cone or Clarke's) appearing elsewhere. The case with a single inequality constraint is discussed in detail by establishing...

Optimal design of mixed AC–DC distribution systems for commercial buildings: A Nonconvex Generalized Benders Decomposition approach

• We develop an MINLP model which determines the optimal design of a mixed AC–DC building electrical distribution system. • We develop a tailored global optimization algorithm, based on Nonconvex Generalized Benders Decomposition. • We provide a case study, including a medium office building. Direct current (DC) electricity distribution systems have been proposed as an alternative to traditional, alternating current (AC) distribution systems for commercial buildings. Partial replacement of AC distribution with DC distribution can improve service to DC loads and overall building energy efficiency. This article develops (i) a mixed-integer, nonlinear, nonconvex mathematical programming problem to determine maximally energy efficient designs for mixed AC–DC electricity distribution systems in commercial buildings, and (ii) describes a tailored global optimization algorithm based on Nonconvex Generalized Benders Decomposition. The results of three case studies demonstrate the strength of the decomposition approach compared to state-of-the-art general-purpose global solvers.