Upper-level Objective Function Research Articles

[formula omitted]-perturbation of bilevel optimization problems: An error bound analysis

In this paper, we analyze a perturbed formulation of bilevel optimization problems, which we refer to as δ-perturbed formulation. The δ-perturbed formulation allows to handle the lower level optimization problem efficiently when there are multiple lower level optimal solutions. By using an appropriate perturbation strategy for the optimistic or pessimistic formulation, one can ensure that the optimization problem at the lower level contains only a single (approximate) optimal solution for any given decision at the upper level. The optimistic or the pessimistic bilevel optimal solution can then be efficiently searched for by algorithms that rely on solving the lower level optimization problem multiple times during the solution search procedure. The δ-perturbed formulation is arrived at by adding the upper level objective function to the lower level objective function after multiplying the upper level objective by a small positive/negative δ. We provide a proof that the δ-perturbed formulation is approximately equivalent to the original optimistic or pessimistic formulation and give an error bound for the approximation. We apply this scheme to a class of algorithms that attempts to solve optimistic and pessimistic variants of bilevel optimization problems by repeatedly solving the lower level optimization problem.

P-facility capacitated location problem with customer-equilibrium decisions: a recreational case study in Mazandaran province

Purpose This paper aims to propose a bi-level mixed integer linear location-allocation problem. The upper-level objective function is dedicated to minimizing the total distances covered by customers to meet the p-selected facilities and the fixed cost values for establishing these facilities. While in the lower level, a customer preference function evaluates the priority of customers in selecting facilities. Design/methodology/approach The solution approach to the proposed model uses the Karush–Kuhn–Tucker (KKT) optimality conditions to the lower-level problem where a set of p-selected facilities are introduced as the selection of the upper-level decision maker. The bi-level model reduces to a single-level model with some added binary variables. Findings Sensitivity analysis of the proposed bi-level model concerning variations of such different parameters as customers’ preferences and the number of selected facilities have been provided, using some numerical examples. Also, locating a recreational facility in Mazandaran province, Iran, has been provided to evaluate the reliability of the proposed model and efficiency of the solution approach, as well. Originality/value To the best of the authors’ knowledge, this paper is original and its findings are not available elsewhere.

Bi-level programming and multi-objective optimization for the distribution of resources in hierarchical organizations

The decision regarding the hierarchical distribution of resources is crucial for many organizations that want to allocate such resources efficiently. At the upper level of the hierarchy, a decision-maker may have an objective and a set of feasible solutions partially determined by the lower level. Nevertheless, the upper level can influence but not control the decision-maker at the lower level. Moreover, the objectives of the upper and lower levels are conflicting. Hence, the hierarchical distribution of resources can be formulated as a bi-level programming model. This paper reformulates the hierarchical allocation of resources. The Hooke-Jeeves (HJ) algorithm and the hybrid scatter search–Nelder-Mead (SSNM) method are proposed to solve the newly formulated problem. The problem is also analyzed from a multi-objective perspective to equally prioritize the upper and lower levels while subjecting each to an interdependent set of constraints; a multi-objective scatter search algorithm (MSSA) was developed for that purpose. Tailor-made instances were also designed for the implementation of the developed algorithms. The findings demonstrated that the HJ algorithm provides the best solution according to the upper-level objective function. However, it does not guarantee the best result for the lower level. In comparison, MSSA has a set of best possible solutions for both objectives. The hybrid SSNM method could not match the results provided by the former methods.

Analysis of strategic bidding of a DER aggregator in energy markets through the Stackelberg game model with the mixed-integer lower-level problem

An open electricity market environment provides sufficient conditions for a distributed energy resource (DER) aggregator to participate in the day-ahead electricity market. Therefore, a one-leader multi-follower Stackelberg game model is developed to analyze the strategic bidding behavior of a DER aggregator and one of the lower-level problems is to check the security of the distribution system (DS), which is elaborated by the mixed-integer second-order cone programming (MISOCP) model considering the active power loss and discrete controls in the DS. Moreover, we first use the Karush–Kuhn–Tucker (KKT)-based reformulation approach to transform the model into a one-leader one-follower model. However, the objective function of the upper-level model cannot be completely linearized by using the strong dual theorem because the active power loss of the DS is considered in the lower-level security check problem. Therefore, we propose a piecewise linear approximation approach based on the data-driven algorithm to linearize the upper-level objective function. Moreover, we apply multivariate linear surrogation to transform this one-leader one-follower model into a single-level MISOCP model. Finally, we verify the correctness of the proposed algorithm by comparing it with the particle swarm optimization (PSO) algorithm in a small integrated transmission and distribution (T&D) system and validate the effectiveness of the proposed algorithm in a practical integrated T&D system.

Energy Aware Trajectory Optimization of Solar Powered AUVs for Optical Underwater Sensor Networks

Visible light communication (VLC) provides an alternative underwater wireless connectivity solution with its low latency and high data rates albeit at relatively shorter distances in the order of tens of meters. In the context of underwater sensor networks (USNs), VLC is particularly suitable to establish connectivity between “data mule” autonomous underwater vehicles (AUVs) and sensor nodes since communications is enabled only when the sensor node and mule AUV are in close proximity. In this paper, we consider a USN scenario where a solar-powered AUV gathers data from the sensor nodes using VLC signaling. We formulate a three-dimensional trajectory optimization for solar-powered AUVs with the goal of maximizing the harvested energy under constraints imposed by the data transmission. The optimization constraints include the minimum required data transfer rate, therefore a corresponding transmission distance, between the sensors and the AUV. We formulate the problem as a bilevel optimization problem. The lower-level objective function is in the form of traveling salesman problem which determines the optimum sequence order of the sensor nodes to be visited while the upper-level objective function is the optimization of the trajectory between each pair of adjacent nodes for the given order of node visits. Our numerical results demonstrate that the proposed trajectory significantly prolongs the mission time and autonomous operation of the AUV without the need to return to home base. Furthermore, since the proposed trajectory optimization is reactive to ocean currents, it brings reductions in the energy consumption of the AUVs.

A deterministic bounding procedure for the global optimization of a bi-level mixed-integer problem

In this paper, a deterministic bounding procedure for the global optimization of a mixed-integer bi-level programming problem is proposed. The aim has been to develop an efficient algorithm to deal with a case study in the electricity retail market. In this problem, an electricity retailer wants to define a time-of-use tariff structure to maximize profits, but he has to take into account the consumers’ reaction by means of re-scheduling appliance operation to minimize costs. The problem has been formulated as a bi-level mixed-integer programming model.The algorithm we propose uses optimal-value-function reformulations based on similar principles as the ones that have been used by other authors, which are adapted to the characteristics of this type of (pricing optimization) problems where no upper (lower) level variables appear in the lower (upper) level constraints. The overall strategy consists of generating a series of convergent upper bounds and lower bounds for the upper-level objective function until the difference between these bounds is below a given threshold. Computational results are presented as well as a comparison with a hybrid approach combining a particle swarm optimization algorithm to deal with the upper-level problem and an exact solver to tackle the lower-level problem, which we have previously developed to address a similar case study. When the lower-level model is difficult, a significant relative MIP gap is unavoidable when solving the algorithm's subproblems. Novel reformulations of those subproblems using “elastic” variables are proposed trying to obtain meaningful lower/upper bounds within an acceptable computational time.

A Novel Methodology to Assimilate Sub-Path Flows in Bi-Level OD Matrix Estimation Process

Traditional bi-level origin-destination (OD) matrix estimation process adjusts the matrix (at the upper level) based on the deviation between the observed and simulated traffic counts. The problem is mathematically under-determined, and the quality of the solution can be enhanced by restricting the upper level search space with information from other sources. This paper presents a methodology that assimilates sub-path flows in the upper level objective function. The contributions of the study are two-fold: first, it proposes the idea of “structural comparison of sub-path flows” to relax the requirement of “known” penetration rate of vehicles’ trajectories; second, it proposes an innovative upper level formulation where the structural difference between the observed and assigned sub-path flows is integrated with the traditional deviations between the observed and assigned link flows. The sub-path flows can be estimated from advanced data sources such as Bluetooth MAC scanner. The proposed methodology is tested using simulation on a realistic network from Brisbane, Australia and results indicate its practical relevance for situations when the penetration rate of Bluetooth trajectories is generally unknown. The proposed method has a better ability to maintain structural consistency and showed considerable improvements in the quality of OD estimates as compared to the traditional traffic counts-based approach.

Research on Energy Storage and High Proportion of Renewable Energy Planning Considering Demand

Based on the high proportion of renewable energy connected to the active distribution network, this article studies the joint planning of demand-side response and energy storage. Firstly, a two-level optimization model is established for the planning of active distribution network. The upper level objective function is the investment, operation and maintenance cost of energy storage and fan, and the lower level objective function is the annual network loss cost. Then, the improved Longhorn algorithm is used to solve the two-layer programming model, and the combination of Longhorn beard algorithm and particle swarm optimization algorithm not only improves the iteration speed, but also improves the global searching ability. Finally, the example analysis proves the effectiveness of joint planning to solve the high proportion of renewable energy, thus improving the economic benefits.

A parametric programming approach to bilevel optimisation with lower-level variables in the upper level

This paper examines linearly constrained bilevel programming problems in which the upper-level objective function depends on both the lower-level primal and dual optimal solutions. We parametrize the lower-level solutions and thereby the upper-level objective function by the upper-level variables and argue that it may be non-convex and even discontinuous. However, when the upper-level objective is affine in the lower-level primal optimal solution, the parametric function is piece-wise linear. We show how this property facilitates the application of parametric programming and demonstrate how the approach allows for decomposition of a separable lower-level problem. When the upper-level objective is bilinear in the lower-level primal and dual optimal solutions, we also provide an exact linearisation method that reduces the bilevel problem to a single-level mixed-integer linear programme (MILP). We assess the performance of the parametric programming approach on two case studies of strategic investment in electricity markets and benchmark against state-of-the-art MILP and non-linear solution methods for bilevel optimisation problems. Preliminary results indicate substantial computational advantages over several standard solvers, especially when the lower-level problem separates into a large number of subproblems. Furthermore, we show that the parametric programming approach succeeds in solving problems to global optimality for which standard methods can fail.

Management of Bilateral Contracts for Gencos Considering the Risk in Spot Market

In electricity market, generation companies (Gencos) face the risk in both the spot market and contract market. Their strategies of contracts in electricity market will directly affect their profits. Therefore, it is crucial for Gencos to determine the total amount of contracts at a certain risk. This paper proposes an optimal management of contract power for generators considering bidding risks in spot market. Based on the risk-based Pareto front, the power quantity-price bidding curve of generators is constructed. We build a bi-level optimization model with upper-level objective function as generators revenue maximization and lower-level total objective function as production cost minimization. The Differential Evolution (DE) technique and Optimal Power Flow (OPF) tools are utilized in model solving. The simulation results of the modified IEEE system show the feasibility and effectiveness of this strategy.

Designing time-of-use tariffs in electricity retail markets using a bi-level model – Estimating bounds when the lower level problem cannot be exactly solved

Time-of-use tariffs are a pricing strategy for a product or service in which the supplier establishes time-differentiated prices. Dynamic (e.g., day-ahead) time-differentiated electricity prices can contribute to increase the retailer's profit, allow end-users to reduce the consumption costs and enhance grid efficiency.The electricity retailer and the consumer are hierarchically related. The interaction between them can be modeled by a bi-level (BL) optimization model – the retailer is the upper level decision maker and the consumer is the lower level decision maker. The retailer and the consumer have different and conflicting goals: the retailer establishes the pricing scheme to sell electricity to consumers to maximize his profit; the consumer reacts to these prices by determining the operation of the controllable loads in order to minimize the discomfort and the electricity bill.In this work, a BL optimization model incorporating shiftable, interruptible and thermostatic loads is proposed. The upper level problem is tackled by a particle swarm optimization algorithm while the lower level problem is solved by an exact mixed-integer programming solver. The inclusion of the thermostatic load in the lower level problem imposes a much higher computational burden. Therefore, it may not be possible to find the optimal lower level solution, and a sub-optimal lower level solution is infeasible to the BL problem. Considering a computational budget, this work proposes an approach to compute good quality estimates of bounds for the upper level objective function, providing the leader further information and allowing him to make sounder decisions in an adequate time frame.

Data Poisoning Attacks on Multi-Task Relationship Learning

Multi-task learning (MTL) is a machine learning paradigm that improves the performance of each task by exploiting useful information contained in multiple related tasks. However, the relatedness of tasks can be exploited by attackers to launch data poisoning attacks, which has been demonstrated a big threat to single-task learning. In this paper, we provide the first study on the vulnerability of MTL. Specifically, we focus on multi-task relationship learning (MTRL) models, a popular subclass of MTL models where task relationships are quantized and are learned directly from training data. We formulate the problem of computing optimal poisoning attacks on MTRL as a bilevel program that is adaptive to arbitrary choice of target tasks and attacking tasks. We propose an efficient algorithm called PATOM for computing optimal attack strategies. PATOM leverages the optimality conditions of the subproblem of MTRL to compute the implicit gradients of the upper level objective function. Experimental results on real-world datasets show that MTRL models are very sensitive to poisoning attacks and the attacker can significantly degrade the performance of target tasks, by either directly poisoning the target tasks or indirectly poisoning the related tasks exploiting the task relatedness. We also found that the tasks being attacked are always strongly correlated, which provides a clue for defending against such attacks.

A modification of the trilevel Kth-Best algorithm

In this paper, we present a modification of the traditional linear trilevel Kth-Best algorithm. The proposed modified Kth-Best algorithm considers the linear trilevel programming problems in which the middle level and the lower level problems are unbounded or their objective functions are inconsistant. These cases are not considered in the trilevel Kth-Best algorithm proposed by Zhang et al. Moreover, we discuss some geometric properties of a linear trilevel programming problem wherein each decision maker might have his (her) own restrictions and the upper level objective function contain lower level variables. Finally, a number of numerical examples are presented and the results are verified as well.

Solving the General Fuzzy Random Bilevel Programming Problem Through <inline-formula> <tex-math notation="LaTeX">$Me$ </tex-math> </inline-formula> Measure-Based Approach

This paper addresses the general fuzzy random bilevel programming problem in which not only both objective functions, but also the constraints contain fuzzy random variable coefficients. From point of view of different decision attitudes of the decision-makers at both levels under the noncooperative case, the expected value based on $Me$ measure is adopted to handle fuzzy stochastic objective functions at the upper and lower levels so as to capture any attitudes between extremely optimistic and pessimistic, and the rough approximation strategy is applied to convert fuzzy stochastic constraints into two approximation stochastic constraints for purpose of avoiding losing much information. Accordingly, the upper and lower approximation stochastic bilevel programming models are built. Then, these two stochastic bilevel programming models are transformed into their equivalent deterministic forms by virtue of expectation optimization and chance constrained conditions. With the help of differential evolution algorithm, the lower and upper bounds of the upper level objective function can be obtained by solving the resulting deterministic problems. By this means, it is helpful to provide more information for decision-makers to select the most desirable alternatives. Finally, a numerical example and a real-case study are provided to demonstrate the effectiveness of the proposed approach.

Seeking the Optimal Solution to Bilevel Linear Programming by Dual Problem

In view of the linear feature of the upper-level and lower-level objective function of the bilevel programming as well as its constraint conditions, this paper presented a global optimization method. Replaced low-level sub-problem of the bilevel linear programming problem with dual problem, and then added the duality gap of the lower-level problem to the upper-level objective function, as a penalty function, and solved single level mathematical programming problem as an alternative to seek the optimal solution of the original problem. Numerical experiments show that the proposed algorithm is effective and feasible. This paper avoided the complexity with solving the dual problem's constraint domain pole, thus saved computing time and energy.

Risk assessment and bi‐level optimization dispatch of virtual power plants considering renewable energy uncertainty

Aiming at the effective management of distributed energy resources (DERs), we propose a bi‐level optimization dispatch model which consists of a virtual power plant (VPP) and an independent system operator (ISO) in order to achieve a coordinated dispatch between the upper level objective function of minimizing system's operation cost and the lower level objective function of maximizing the VPP's net profit. However, the VPP's dispatch strategy is variable due to the threat from unavoidable uncertainties including unit outage, load forecast deviation, and renewable energy forecasting error. In order to evaluate the influences of these risks on VPP's dispatch strategy, a proper definition of expected energy not supplied (EENS) and its relationship with VPP's reserve capacity are provided. Also, chance‐constrained programming (CCP) is adopted during the analysis. By including VPP's risk cost into the dispatch model, VPP's dispatch strategy is able to keep a balance between the potential risk and economic profit. In addition, a further discussion regarding the risk cost is conducted in view of renewable energy capacity proportion and the VPP capacity proportion. Based on the analysis results of the case study provided, the proposed model serves as a foundation for VPP's economic and risk‐balanced dispatch.

A heuristic algorithm solving bilevel toll optimization problems

Purpose – One of the most important problems concerning the toll roads is the setting of an appropriate cost for traveling through private arcs of a transportation network. The purpose of this paper is to consider this problem by stating it as a bilevel programming (BLP) model. At the upper level, one has a public regulator or a private company that manages the toll roads seeking to increase its profits. At the lower level, several companies-users try to satisfy the existing demand for transportation of goods and/or passengers, and simultaneously, to select the routes so as to minimize their travel costs. In other words, what is sought is kind of a balance of costs that bring the highest profit to the regulating company (the upper level) and are still attractive enough to the users (the lower level). Design/methodology/approach – With the aim of providing a solution to the BLP problem in question, a direct algorithm based on sensitivity analysis (SA) is proposed. In order to make it easier to move (if necessary) from a local maximum of the upper level objective function to another, the well-known “filled function (FF)” method is used. Findings – The paper proposes and tests two versions of the heuristic algorithm to solve the toll optimization problem (TOP) based upon SA for linear programming (LP) problems. The algorithm makes use of an SA procedure for the LP problem at the lower level, as well as of the “filled” function technicalities in order to reach the global optimum when “jammed” at some local optimum. Numerical experiments with a series of small and medium dimension test problems show the proposed algorithm’s robustness and decent convergence characteristics. Practical implications – Numerical experiments with a series of small- and medium dimension test problems show the proposed algorithm’s robustness and reasonable convergence characteristics. In particular, while ceding in efficiency to other algorithms when solving small problems, the proposed method wins in the case of medium (higher dimensional) test models. Because of that, one can expect a serious real-life impact on the TOP when the proposed methods and/or their improved versions are developed further to be applicable in practice in the near future. Originality/value – The proposed algorithms are original and perform well when solving small and medium test numerical problems. The proposed heuristics aim at filling in a gap in a series of numerical approaches to the solution of TOP problem listed in the Introduction. To the authors knowledge, no systematic attempts to apply the SA tools to the toll assigned problem have been recently made. Moreover, the combination of these powerful tools with the “FFs” techniques brings forward some new global optimization ideas. Exactly these features build up the knowledge this specific paper offers in relation to previous relevant works.

A matheuristic for the discrete bilevel problem with multiple objectives at the lower level

AbstractIn this paper, we solve a discrete bilevel problem with multiple objectives at the lower level and constraints at the upper level coupling variables of both levels. In the case of a multiobjective lower level, we deal with a set of Pareto‐efficient solutions rather than a single optimal lower level solution. To calculate the upper level objective function value, we need to select one solution out of a potentially large set of efficient lower level solutions. To avoid the enumeration of the whole set of Pareto solutions, we formulate an auxiliary mixed integer linear programming problem with a large number of constraints. We propose an iterative exact method to solve it. To find a near‐optimal upper level solution, we apply a metaheuristic. The method is tested on the discrete ()‐centroid problem with multiple objectives at the lower level.

A bilevel formulation of the pediatric vaccine pricing problem

We consider the characterization of optimal pricing strategies for a pediatric vaccine manufacturing firm operating in an oligopolistic market. The pediatric vaccine pricing problem (PVPP) is formulated as a bilevel mathematical program wherein the upper level models a firm that selects profit-maximizing vaccine prices while the lower level models a representative customer’s vaccine purchasing decision to satisfy a given, recommended childhood immunization schedule (RCIS) at overall minimum cost. Complicating features of the bilevel program include the bilinear nature of the upper-level objective function and the binary nature of the lower-level decision variables. We develop and test variants of three heuristics to identify the pricing scheme that will maximize a manufacturer’s profit: a Latin Hypercube Sampling (LHS) of the upper-level feasible region, an LHS enhanced by a Nelder–Meade search from each price point, and an LHS enhanced by a custom implementation of the Cyclic Coordinate Method from each price point. The practicality of the PVPP is demonstrated via application to the analysis of the 2014 United States pediatric vaccine private sector market. Testing results indicate that a robust sampling method combined with local search is the superlative solution method among those examined and, in the current market, that a manufacturer acting unilaterally has the potential to increase profit per child completing the RCIS by 35 percent (from 231.84 to 312.55 dollars) for GlaxoSmithKline, 47 percent (from 63.96 to 93.70 dollars) for Merck, and 866 percent (from 25.99 to 251.04 dollars) for Sanofi Pasteur over that obtained via current pricing mechanisms.

A New Method To Solve Bi-Level Quadratic Linear Fractional Programming Problems

In this paper, we have developed a new method to solve bi-level quadratic linear fractional programming (BLQLFP) problems in which the upper-level objective function is quadratic and the lower-level objective function is linear fractional. In this method a BLQLFP problem is transformed into an equivalent single-level quadratic programming (QP) problem with linear constraints by forcing the duality gap of the lower-level problem to zero. Then by obtaining all vertices of the constraint region of the dual of the lower-level problem, which is a convex polyhedron, the single-level QP problem is converted into a series of finite number of QP problems with linear constraints which can be solved by any standard method for solving a QP. The best among the optimal solutions gives the desired optimal solution for the original bi-level programming (BLP) problem. Theoretical results have been illustrated with the help of a numerical example.