Interval Quadratic Programming Research Articles

A recurrent neural network approach for nonconvex interval quadratic programming

In the paper, a new approach is presented to tackle an interval optimization problem with nonconvex interval objective function and affine interval equality and inequality constraints utilizing a recurrent neural network. The approach involves converting the nonconvex interval quadratic programming into a cubic optimization via convex combinations and considering its weight as a decision variable. The recurrent neural network is then established to solve the cubic optimization through subgradient techniques and an exterior penalty function method. Moreover, it is proven that there is a unique global solution to the recurrent neural network, and its trajectory can reach the feasible region in a finite time and remain there. Additionally, it is demonstrated that the trajectory of recurrent neural network converges exponentially or in finite time towards a singleton belonging to the set of critical points of the cubic optimization. This approach is beneficial for interval quadratic programming problems where the weight is unknown or challenging to determine in practical applications. Finally, the effectiveness of this approach is verified through two numerical examples.

Coupling optimization of irrigation and fertilizer for synergic development of economy-resource-environment: A generalized inexact quadratic multi-objective programming

Water and nutrients are two of the most important factors affecting agricultural production in arid regions, however, the misuse of agricultural chemicals has led to serious pollution. In order to achieve synergic regional economy-resource-environment development, this study proposes a generalized interval quadratic multi-objective programming model with fuzzy-interval credibility constraints (FIC-GIQMP). By introducing interval water-fertilizer production function, the quantitative relationship between crop yield, and evapotranspiration-fertilizer can be accurately reflected. The model can seek a balance among economic growth, resources security, and environmental pollution reduction, as well as address complexities, uncertainties, and risks by water-fertilizer management. Moreover, a modified duality-based algorithm is introduced to overcome the difficulty of solving the FIC-GIQMP model. The developed framework was applied to Shiyang River Basin in Northwest China. Optimal water and fertilizer allocations among different crops within each period under different scenarios were generated. Compared to the status quo, the optimal net benefit increased by 17.89%, while the carbon and water footprints decreased by 47.99% and 32.11%, respectively. Apart from giving the coupled water-fertilizer allocation schemes, the optimal results further appeal for decision-makers to modify irrigation pattern and crop structures.

Data-driven interval credibility constrained quadratic programming model for water quality management under uncertainty

Although integrated simulation-optimization modeling can provide a comprehensive and reliable analysis for water quality management (WQM), it is usually not easy to implement in practice. This study proposed a new efficient simulation-optimization modeling approach by leveraging the power of data-driven modeling, to support WQM under various uncertainties. A water quality simulation model is integrated with the optimization model, and then substituted by a series of numerical surrogate models based on inexact linear regression. The transformation can significantly reduce the computational burden and make it possible to implement uncertainty quantification through hybrid inexact programming. The proposed model incorporates interval quadratic programming and credibility constrained programming to deal with nonlinearity and various uncertainties associated with the management system. The proposed approach is applied to a real case study of the Grand River watershed in Canada for controlling phosphorus concentration in river water. The Grand River Simulation Model (GRSM) is employed as the physical simulation model to estimate the total phosphorus concentration in the river. Interval solutions under different confidence levels of violating the effluent standards were obtained, which can be used to generate optimal phosphorus control strategies. The results indicate the proposed data-driven interval credibility constrained quadratic programming (DICCQP) model is able to provide reliable and robust solutions for WQM by considering nonlinearity and various uncertainties while maintaining a high computational efficiency. The proposed new framework can be extended and applied to the other watersheds. The high efficiency of the proposed model makes it possible to solve large-scale complex water quality management and planning problems.

Application of Stochastic Programming Technique to Solve Interval Quadratic Programming Problem

In this paper we developed a methodology to solve interval quadratic programming problem (IQPP). In the proposed work we convert interval quadratic programming problem (IQPP) into a non-interval stochastic programming problem. We transform the interval valued objective function into function of random variables and corresponding equality/inequality interval valued constraints into stochastic constraints using Gaussian distribution and 6σ-rule. Using this transformation method we tried to develop a solution of native interval quadratic programming problem. The methodologies are illustrated with numerical examples.

Solving Interval Quadratic Programming Problems by Using the Numerical Method and Swarm Algorithms

In this paper, we present a new approach which is based on using numerical solutions and swarm algorithms (SAs) to solve the interval quadratic programming problem (IQPP). We use numerical solutions for SA to improve its performance. Our approach replaced all intervals in IQPP by additional variables. This new form is called the modified quadratic programming problem (MQPP). The Karush–Kuhn–Tucker (KKT) conditions for MQPP are obtained and solved by the numerical method to get solutions. These solutions are functions in the additional variables. Also, they provide the boundaries of the basic variables which are used as a start point for SAs. Chaotic particle swarm optimization (CPSO) and chaotic firefly algorithm (CFA) are presented. In addition, we use the solution of dual MQPP to improve the behavior and as a stopping criterion for SAs. Finally, the comparison and relations between numerical solutions and SAs are shown in some well-known examples.

Simulation-based interval chance-constrained quadratic programming model for water quality management: A case study of the central Grand River in Ontario, Canada

Effective river water quality management and planning is a complex issue challenged by various complexities and uncertainties. A simulation-based interval chance-constrained quadratic programming (ICCQP) model is developed for the seasonal planning of water quality management (WQM) under various uncertainties. The proposed model incorporates interval quadratic programming, chance-constrained programming, and a seasonal water quality simulation model within a general framework for WQM. Uncertainties associated with the objective and the coefficients in the left-hand sides of the constraints are tackled as intervals. Meanwhile, parameter uncertainties on the right-hand sides are characterized using probability distributions. Nonlinearities in the cost function are reflected by quadratic programming. A multi-segment water quality model is used to simulate the dynamic interactions between wastewater discharges and river water quality. The proposed ICCQP-WQM model is applied in a real case study for the control of total phosphorus (TP) in the central Grand River in Ontario, Canada. The results demonstrate that the proposed model is able to incorporate uncertainties expressed as intervals and probability information into an optimization framework and provide interval solutions. Thus, different cost-effective schemes for seasonal WQM could be generated. The results show the Kitchener wastewater treatment plant (WWTP) affects the value of the objective function more than the other WWTPs in the study area. It is also found that the Kitchener WWTP's cost accounts for the highest proportion (approximately 35.1–37.9%) of the total annual cost, which implies the control of TP at the Kitchener plant is the most important to the system. Moreover, river water TP standards in spring and autumn are usually difficult to meet, indicating different TP control strategies are needed in these two seasons. The generated results are valuable for local decision makers to generate TP control strategies, and also to identify optimized solutions under various uncertainties. The proposed ICCQP-WQM model can be extended to other watersheds to support effective water quality management and planning.

Development of an interval quadratic programming water quality management model and its solution algorithms

Interval quadratic programming (IQP) is one of the most popular programming techniques for addressing the uncertainties associated with nonlinear environmental management problems. However, efficient algorithms for solving IQP problems in water quality management (WQM) have not been well studied. In this study, an IQP model is developed for WQM under uncertainty. Three solution algorithms, including a piecewise linear approximation (PLA) method, a derivative algorithm (DEA) and a duality-based algorithm (DUA) are proposed for solving the IQP-WQM problem. The developed model and the corresponding solution algorithms are applied to a hypothetic WQM problem to demonstrate their applicability. The results show the lower bounds of the total cost obtained by three algorithms have a relationship of fDUA−=fDEA−≤fPLA−, while that of the upper bounds is fDEA+≤fPLA+≤fDUA+. Moreover, the sensitivity analysis shows that no matter how the IQP-WQM model is solved, the model response from the three solution algorithms is consistent. The results indicate that all of the three algorithms can efficiently deal with quadratic programming problems under uncertainties in the format of intervals. Comparison among the three algorithms shows that DUA provides interval solutions with wider ranges than the other two methods, and it requires less computational efforts than DEA. It is also found that PLA is more flexible and might require lower computational efforts for large scale problems. This study can provide useful technical support for effective WQM.

Optimal solution set in interval quadratic programming problem

There are several methods to compute the optimal bounds of the objective function for interval quadratic programming (IQP) problems, but no method has yet been suggested to calculate a set of optimal solutions of IQP problems. This paper presents an accurate set of optimal solutions for the interval quadratic programming problems. The optimal solution of the quadratic programming problem is not essentially an extreme point. We rst propose conditions that make the optimal solutions of the IQP to extreme points and then, using these conditions, we compute the exact set of optimal solutions for the IQP problem.

Numerical solution of a general interval quadratic programming model for portfolio selection.

Based on the Markowitz mean variance model, this paper discusses the portfolio selection problem in an uncertain environment. To construct a more realistic and optimized model, in this paper, a new general interval quadratic programming model for portfolio selection is established by introducing the linear transaction costs and liquidity of the securities market. Regarding the estimation for the new model, we propose an effective numerical solution method based on the Lagrange theorem and duality theory, which can obtain the effective upper and lower bounds of the objective function of the model. In addition, the proposed method is illustrated with two examples, and the results show that the proposed method is better and more feasible than the commonly used portfolio selection method.

An Interval Quadratic Fuzzy Dependent-Chance Programming Model for Optimal Irrigation Water Allocation under Uncertainty

In this study, an interval quadratic fuzzy dependent-chance programming (IQFDCP) model is proposed for irrigation water management under uncertainty. This model can deal with the interval and fuzzy uncertainties encountered in optimal irrigation water resources allocation problems. It takes optimal credibility level of system revenue as the objective function for addressing the conflict between maximum system revenue and the credibility level of the system revenue. By providing three scenarios of precipitation, the optimal solutions are obtained. The developed model is applied to a case study for irrigation water allocation in Minqin Oasis, Wuwei city, northwest China. This study can help determine how much irrigation water should be allocated to different crops under the maximum credibility level of the system revenue. The optimal solutions can provide a certain credibility level of system revenue corresponding to different scenarios and mitigate the system-failure risk level of water shortages in arid areas. Moreover, because the IQFDCP can objectively reflect the interrelationship among crop irrigation water amount, system revenue, and system-failure risk level, the results are able to provide advices to decision makers for efficiently managing water resources in different scenarios.

Applying norm concepts for solving interval support vector machine

Abstract In this paper, a new variation of Support Vector Machine (SVM) is proposed which is named Interval Support Vector Machine (ISVM). The uncertainty of input data is one of the main problems to handle the real-world problems. Uncertainty factors could be a result of random variables existence, incorrect or imperfect data, and approximations instead of measurements or incomparability of data that is very dependent on different measurement or observation conditions. Interval numbers generally can be used for the representation of real data. In this version, SVM with interval samples is presented and reformulation of SVM performs real sample classification. Here, ISVM could be interchanged with an interval quadratic programming problem which itself can be converted to a pair of the so-called two-level mathematical programs based on norm concepts. The two-level mathematical programs are formed to find the upper and the lower bounds corresponding to the objective values of the interval quadratic program. Numerical experiments are examined for variant datasets such as normal, noisy, and interval-valued dataset to illustrate the performance of our approach.

Some Properties of Interval Quadratic Programming Problem

For interval liner programming problems, Rohn proposed four equivalence relations regarding to the upper and lower bounds of the interval optimal value. In this paper, similar problems of interval quadratic programming problem have been discussed. Some interesting properties have been proved and an illustrative example and remarks are given to get an insight of the properties.

Some properties of the lower bound of optimal values in interval convex quadratic programming

One of the fundamental problems in interval quadratic programming is to compute the range of optimal values. In this paper, we derive some results on the lower bound of interval convex quadratic programming. We first develop complementary slackness conditions of a quadratic program and its Dorn dual. Then, some interesting and useful characteristics of the lower bound of interval quadratic programming are established based on these conditions. Finally, illustrative examples and remarks are given to get an insight into the problem discussed.

Existence of Equilibrium Points for Bimatrix Game with Interval Payoffs

This paper deals a bimatrix game with payoffs as closed intervals. Existence of equilibrium point of this game is discussed by using suitable interval quadratic programming problem. Further, a methodology is proposed for finding optimal strategies for each player of the game. The methodology is illustrated by numerical example.

Some results on the upper bound of optimal values in interval convex quadratic programming

One of the fundamental problems in interval quadratic programming is to compute the range of optimal values. For minimized problem with equality constraint, computing the upper bound of the optimal values is known to be NP-hard. One kind of the effective methods for computing the upper bound of interval quadratic programming is so called dual method, based on the dual property of the problem. To obtain the exact upper bound, the dual methods require that the duality gap is zero. However, it is not an easy task to check whether this condition is true when the data may vary inside intervals. In this paper, we first present an easy and efficient method for checking the zero duality gap. Then some relations between the exact upper bound and the optimal value of the dual model considered in dual methods are discussed in detail. We also report some numerical results and remarks to give an insight into the dual method’s behavior.

Interval quadratic programming for day-ahead dispatch of uncertain predicted demand

In this paper, we propose an interval quadratic programming method for the day-ahead scheduling of power generation and battery charge cycles, where the prediction uncertainty of power consumption and photovoltaic power generation is described as a parameter vector lying in an interval box. The interval quadratic programming is formulated as the problem of finding the tightest box, i.e., interval hull, that encloses the image of a function of the minimizer in parametric quadratic programming. To solve this problem in a computationally efficient manner, we take a novel approach based on a monotonicity analysis of the minimizer in the parametric quadratic programming. In particular, giving a tractable parameterization of the minimizer on the basis of the Karush–Kuhn–Tucker condition, we show that the monotonicity analysis with respect to the parameter vector can be relaxed to the sign pattern analysis of an oblique projection matrix. The monotonicity of the minimizer is found to be essential in the day-ahead dispatch problem, where uncertain predicted demand, described by a parameter vector, is dispatched to power generation and battery charge cycles while the economic cost is minimized. Finally, we verify the efficiency of the proposed method numerically, using experimental and predicted data for power consumption and photovoltaic power generation.

Solving non-linear portfolio optimization problems with interval analysis

Estimation errors or uncertainities in expected return and risk measures create difficulties for portfolio optimization. The literature deals with the uncertainty using stochastic, fuzzy or probability programming. This paper proposes a new approach to treating uncertainty. By assuming that the expected return and risk vary within a bounded interval, this paper uses interval analysis to extend the classical mean-variance portfolio optimization problem to the cases with bounded uncertainty. To solve the interval quadratic programming problem, the paper adopts order relations to transform the uncertain programme into a deterministic programme, and includes the investors’ risk preference into the model. Numerical analysis illustrates the advantage of this new approach against conventional methods.

New method for computing the upper bound of optimal value in interval quadratic program

We consider the interval quadratic programming problems. The aim of this paper is to present a new method to compute the upper bound of the optimal values, under weaker conditions. Moreover, we discuss the relations between the new method and previous results. The features of the proposed methods are illustrated by some examples.

Planning of Optimal Daily Power Generation Tolerating Prediction Uncertainty of Demand and Photovoltaics

The concern with renewable energy has been growing. Large-scale installation of photovoltaic (PV) generation and electricity storage is expected to be installed into the power system in Japan. In this situation, we need to keep supply-demand balance by systematically using not only traditional power generation systems but also the PV generation and storage equipment. Towards this balancing, a number of prediction methods for PV generation and demand have been developed in literature. However, prediction-based balancing is not necessarily easy. This is because the prediction of PV generation and the demand forecasting inevitably includes some uncertainty. Against this background, we formulate a problem to plan battery charge pattern while minimizing the fuel cost of generators with explicit consideration of prediction uncertainty. In this problem, given as interval quadratic programming, the prediction uncertainty is described as a parameter in constraint condition. Furthermore, we propose a method to find a solution to this problem from the viewpoint of monotonicity analysis. Finally, by numerical analysis based on this problem and its solution method, we discuss the relation between the minimal regulating capacity and the required battery charge/discharge pattern to tolerate a given amount of prediction uncertainty.

Inexact quadratic joint-probabilistic programming for water quality management under uncertainty in the Xiangxi River, China

An inexact quadratic joint-probabilistic programming model for water quality management (IQJWQ) is developed and applied to supporting multiple-point-source waste reduction in the Xiangxi River, China. The IQJWQ is a hybrid of interval quadratic programming, joint probabilistic programming and multi-segment water quality simulation. It has advantages in reflecting uncertainties expressed as joint probabilities of system risk, probability distributions of water quality standards, interval parameters and nonlinearities in the objective function. An interactive and derivative algorithm is employed for solving the IQJWQ model. The results indicate that the Pingyikou chemical plant and Liucaopo chemical plant contribute more to pollution of the main stream in the Xiangxi River, which should be the prior plants to reduce the wastewater discharge and enhance the wastewater treatment efficiencies. Meanwhile, the environmental agencies should choose the joint probability carefully to balance the tradeoff between production development and pollution control. Compared with the conventional chance-constrained programming method, the IQJWQ exhibits an increased robustness in handling the overall system risk in the optimization process. Although this study is the first application of the IQJWQ to water quality management, the proposed methods in the IQJWQ can also be applicable to many other environmental management problems under uncertainty.