Parametric Programming Problem Research Articles

Fast quadratic model predictive control based on sensitivity analysis and Wolfe method

AbstractThis paper proposes a new algorithm based on sensitivity analysis and the Wolfe method to solve a sequence of parametric quadratic programming (QP) problems such as those that arise in quadratic model predictive control (QMPC). The Wolfe method, based on Karush–Kuhn–Tucker conditions, has been used to convert parametric QP problems to parametric linear programming (LP) problems, and then the sensitivity analysis is applied to solve the sequence of the parametric LP problems. This strategy obtains sensitivity analysis‐based QMPC (SA‐QMPC) algorithm. It is proved that the computational complexity of SA‐QMPC is for a region of the initial conditions and for sufficiently small sampling time and all initial conditions, where and are the horizon time and dimension of the state vector, respectively. Numerical results indicate the potential and properties of the proposed algorithm.

Multi-Stage Mathematical Programming Models and Their Applications in Agriculture

The article presents multi-stage parametric programming models, the characteristics of which depend on time and seasonality. The first model took into account the impact on the yield of agricultural crops of predecessors. In the second model, related to the distribution of sales of products by seasons, at the first stage, a point and interval forecast of the task parameters is carried out, at the second stage, optimal coefficients are determined that characterize the volume of sales of crop products depending on the season. Then, based on the obtained forecasts and optimal coefficients, the parametric programming problem is solved. The described tasks are implemented on real objects of the Irkutsk region. Logistic and asymptotic dependences are used to predict crop yields, and the price trend is a seasonal model. It should be noted that the presented models can be refined with the help of expert assessments, which will improve the adequacy of planning the activities of agricultural producers.

Optimality conditions and sensitivity analysis in parametric convex minimax programming

ABSTRACT In this paper, we study optimality conditions as well as sensitivity analysis of parametric convex minimax programming problems. The main tools used here are the formulas for computing the subdifferentials of maximum functions under some suitable regularity conditions. More precisely, we use these tools to study optimality conditions for the problem under consideration. These results are then applied to obtain optimality conditions for multiobjective optimization problems, which constitute the first part of the paper. In the second part, we derive formulas for computing the subdifferential in the sense of convex analysis and singular subdifferential of the optimal value function for the problem in question.

A digital economy development index based on an improved hierarchical data envelopment analysis approach

The digital economy is playing an increasingly important role in the global economy. National and international organizations commonly utilize a composite index composed of multi-dimensional indicators to monitor performance, analyse policies, and communicate in the digital economy. This study introduces a hierarchical framework for constructing a Digital Economy Development Index (DEDI). One of the key challenges is determining the attribute weights to be assigned by aggregating the sub-indicators with hierarchical dimensions for DEDI. The current study shows that an H-DEA model can be transformed into a parametric linear programming problem and develops an improved golden section algorithm to search for approximate optimal solutions. The newly developed method is highly robust and provides an alternative procedure to determine the optimal weights for building a DEDI. We established a hierarchical evaluation framework and utilized a new approach to measure the DEDI of 30 provinces in China from 2015 to 2020. The results show that inter-provincial digital economy development in China shows a sequential weakening from the east, centre, and northeast to the west. The East leads the country's digital economy overall, while there is a clear gradient gap between provinces. The central region needs to create a cluster for the development of digital industries. The Northeast should enhance the competitiveness of the industry across the board. The West must strengthen the construction of innovative talent and new infrastructure for the digital economy. These findings can serve as guidelines for designing China's ongoing digital economy development.

Robust Parametric Programming for Adaptive Piecewise Linear Control of Photovoltaic Inverters to Regulate Voltages in Power Distribution Systems

This paper proposes a robust parametric programming (RPP) method to adaptively obtain piecewise linear control functions of photovoltaic (PV) inverters for real-time voltage regulation in distribution systems. First, the voltage regulation problem is designed as a multi-parameter programming problem, which is transformed into single-parameter linear programming (sp-LP) problems for realizing local control of a PV inverter while considering the maximum voltage deviation caused by other PV units. Second, an alternating iterative algorithm is developed to solve the sp-LP problem of each PV unit and obtain the control functions of all the PV units. Third, considering the information on PV power correlation, a two-level parametric programming problem is established to enhance the voltage regulation performance of PV inverters. A geometric method is developed to solve the inner-level problem, and breakpoint reduction with robust interpolation is developed to improve the computational efficiency in solving the outer-level problem. Four distribution systems integrated with PV units are used to validate the effectiveness of the proposed method. Numerical results show that the performance of the proposed PV inverter control strategy is close to that of the ideal optimal solution and superior to existing piecewise linear control strategies.

DIRECTIONAL DIFFERENTIABILITY OF THE OPTIMAL VALUE IN QUADRATIC PROGRAMMING PROBLEMS UNDER LINEAR CONSTRAINTS ON HILBERT SPACES

We investigate the first-order directional differentiability of the optimal value function in parametric quadratic programming problems under linear constraints in Hilbert spaces. We derive an explicit formula for computing the directional derivative of the optimal value function in cases where the quadratic part of the objective function is in Legendre form.

A Short Literature on Linear Programming Problem

Researchers and scientists have developed various approaches and methodologies over time to model and analyze different types of linear programming problems, such as assignment problems and parametric programming problems. This paper provides a critical review and classification of existing modelling approaches and solution methods related to linear programming problems. Moreover, the simplex method is discussed in detail through a comprehensive literature review. The paper concludes by presenting an integrated research framework that is directly applicable to the present context, along with suggestions for future research directions.

Использование функций обратных связей для решения задач параметрического программирования

Использование функций обратных связей для решения задач параметрического программирования

Operation Optimization Adapting to Active Distribution Networks Under Three-Phase Unbalanced Conditions: Parametric Linear Relaxation Programming

Asymmetric phase distributed generation, which is widely used in distribution systems, aggravates the severity of the intrinsic three-phase unbalanced condition of distribution systems. To overcome these problems, operation optimization formulas of the three-phase unbalanced active distribution network (ADN) are established considering soft open point (SOP) and belong to a nonconvex quadratic programming (NQP) problem. However, in conventional methods, either approximations that are not valid for an unbalanced distribution system are used or convex relaxation methods are applied to primitive nonlinear optimization models. However, these methods do not guarantee a feasible optimization solution. Therefore, determining the solution to the NQP problem is challenging. In this article, we propose a parametric linear relaxation algorithm to obtain the optimal feasible solution by solving the operation optimization problem of the three-phase unbalanced ADN with the SOP. The parametric linear relaxation technique is constructed based on the properties of quadratic functions and the mean value theorem, and the NQP is converted to a series of parametric linear relaxation programming problems. The results revealed that the proposed algorithm can achieve the optimal and feasible solution and considerably reduce the computation time compared with an equivalent NLP optimization model for the same distribution system.

Mean–variance value at risk criterion for solving a p-median location problem on networks with type-2 intuitionistic fuzzy weights

In this paper, we consider the p-median location problem on networks in which the vertex weights are the type-2 intuitionistic fuzzy variables. First, the credibilistic value at risk of the intuitionistic fuzzy variables is introduced. Using value at risk criterion, we define a new method for reducing of type-2 intuitionistic fuzzy variables to intuitionistic fuzzy variables that is named the value at risk reduced intuitionistic fuzzy variables. Further, the mean and variance values of value at risk reduced intuitionistic fuzzy variables are described. Finally, we apply the mean–variance criterion for solving the type-2 intuitionistic fuzzy p-median location problem. Also, it is shown that this problem is equivalent to a parametric quadratic convex programming problem.

Sub-6 GHz V2X-assisted MmWave optimal scheduling for vehicular networks

Sub-6 GHz assisted millimeter wave (mmWave) technology is a promising solution to address the ever-increasing data exchange demand in Vehicle-to-Everything (V2X) communication. The omni-directional sub-6 GHz channels exchange vehicle and control information in the vehicular network, while the directional mmWave technology is used for high-rate data transmission. Consider the huge amounts of vehicular data, the transmission constraints imposed by mmWave, and the unique vehicular ad hoc network (VANET) data features, a sub-6 GHz assisted-mmWave scheduling algorithm is needed to facilitate V2X data transmission. Existing scheduling algorithms in the literature fall short of addressing realistic V2X data features. Therefore, in this paper, we propose a novel objective to maximize the network utility that succinctly captures realistic V2X data features, including varying data sizes, data importance and their freshness decay with transmission delay. We formulate the scheduling problem into a novel mixed-integer sum-of-ratios optimization problem. To solve the challenging NP-complete problem, we first derive an equivalent parametric mixed-integer linear programming problem with existence and uniqueness proofs. Next, we provide the necessary condition for the optimality of the equivalent problem. A novel Parameterization-based Iterative Algorithm (PIA) is then developed to learn the global optimal solution with convergence analysis. Comparative simulation studies with the brute-force algorithm, the widely used branch and bound (BB) method, the greedy algorithm, and our previously developed maximum weight matching based iterative algorithm (MWMIA) are conducted to verify the correctness and effectiveness of the proposed PIA scheduling algorithm.

A dynamic optimal solution approach for solving neutrosophic transportation problem

Neutrosophic Set (NS) allows us to handle uncertainty and indeterminacy of the data. Several researchers have investigated the Transportation Problems (TP) with various forms of input data. This paper emphasizes a dynamic optimal solution framework for TPs in a neutrosophic setting. This paper investigates a Neutrosophic Transportation Problem (NTP) in which supply, demand, and transportation cost are considered as Single-Valued Neutrosophic Trapezoidal Numbers (SVNTrNs). The weighted possibilistic mean value of their truth, indeterminacy, and facility membership function are calculated. Then, NTP is modelled as a parametric Linear Programming Problem (LPP) and solved. Further, the drawbacks of the existing approaches and advantages of the developed method are discussed. Finally, the real-time problem and numerical illustrations are presented and compared to existing solutions. This study helps the Decision-Makers (DMs) in budgeting their transportation expenses through strategic distribution.

Optimal Control Policy for a Two-Phase M/M/1 Unreliable Gated Queue under N-Policy with a Fuzzy Environment

The two-phase service models analyzed by several authors considered only the probabilistic nature of the queue parameters with fixed cost elements. But the queue parameters and cost elements will be in general are of both possibilistic and probabilistic in nature. Analyzing the performance of the queueing systems with fuzzy environment facilitates to investigate for the possibilistic interval estimates to the performance measures of a queueing system rather than point estimates. In this work, it is proposed to construct membership function of the fuzzy cost function to obtain confidence estimates for some performance measures of a controllable two-phase service single server Markovian gated queue with server startups and breakdowns under N-policy in which the queue parameters viz. arrival rate, startup rate, batch service rate, individual service rate, repair rate and cost elements are all defined as fuzzy numbers. Based on Zadeh’s extension principle and the α-cuts, a set of parametric nonlinear programming problems are developed to find the upper and lower bounds of the minimum total expected cost per unit time at the possibility level α. As the analytical solutions of the nonlinear programming problems developed for the proposed model are tedious, considering the system parameters and cost elements as trapezoidal fuzzy numbers, numerical results for the lower and upper bounds of the optimal threshold N* and the minimum total expected cost per unit time are computed using the nonlinear programming solver available in MATLAB.

Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem

<p style='text-indent:20px;'>This paper deals with the generalized Clarke epiderivative of the extremum multifunction of a multi-objective parametric convex discrete optimal control problem with linear state equations and control constraints. By establishing an abstract result on the generalized epiderivative of the extremum multifunction of a multi-objective parametric convex mathematical programming problem, we derive a formula for computing the generalized Clarke epiderivative of the extremum multifunction to a multi-objective parametric convex discrete optimal control problem. Examples are given to illustrate the obtained results.</p>

A kernel path algorithm for general parametric quadratic programming problem

It is well known that the performance of a kernel method highly depends on the choice of kernel parameter. A kernel path provides a compact representation of all optimal solutions, which can be used to choose the optimal value of kernel parameter along with cross validation (CV) method. However, none of these existing kernel path algorithms provides a unified implementation to various learning problems. To fill this gap, in this paper, we first study a general parametric quadratic programming (PQP) problem that can be instantiated to an extensive number of learning problems. Then we provide a generalized kernel path (GKP) for the general PQP problem. Furthermore, we analyze the iteration complexity and computational complexity of GKP. Extensive experimental results on various benchmark datasets not only confirm the identity of GKP with several existing kernel path algorithms, but also show that our GKP is superior to the existing kernel path algorithms in terms of generalization and robustness.

Sensitivity Analysis of Multi-objective Optimal Control Problems

Sensitivity analysis of multi-objective parametric optimal control problems with nonconvex cost functions and control constraints is given in this paper. By establishing an abstract result on the Mordukhovich subdifferential of the efficient point multifunction of a multi-objective parametric mathematical programming problem, we derive a formula for computing the Mordukhovich subdifferential of the efficient point multifunction to a multi-objective parametric optimal control problem.

Ranking ranges in cross-efficiency evaluations: A metaheuristic approach

In the cross-efficiency evaluation, the existence of alternative optima for weights represents an inconvenience because the ranking obtained depends on the weights used. Alcaraz et al. (2013) design an algorithm that provides a range of positions for each unit, taking into account the entire spectrum of feasible weights that configure the optimal alternative solutions in DEA evaluation. This procedure involves nonlinear models that are solved using parametric mixed-integer non-linear programming problems that can become very complex and laborious and often makes it difficult to obtain the optimum. In this paper, ranking range is approached using metaheuristic techniques which, when well designed, are able to achieve good approximations and they are the preferred alternative when exact techniques require excessive computational effort or they are not able to provide an optimal solution. Some examples widely used in the context of cross-evaluation are used to verify the suitability of the proposed method.

Differential stability of discrete optimal control problems with mixed contraints

Differential stability of parametric discrete optimal control problems in Banach spaces with nonconvex cost functions, nonlinear state equations and mixed constraints is studied. By establishing an abstract result on the subdifferentials of the value function of a parametric mathematical programming problem with geometrical, equality and inequality constraints, we derive formulas for computing the Mordukhovich subdifferential and the singular subdifferential of the value function to a parametric discrete optimal control problem.

Energy-Efficient Power Allocation for D2D Communication underlaying Cellular Networks

This paper considers the power allocation problem in device-to-device (D2D) communication underlaying a cellular network and investigates the impact of different transmitting and interference power constraints on the energy efficiency and spectral efficiency of the network. We formulate the power allocation problem in D2D communication as a nonlinear fractional programming problem with an objective to maximize the energy efficiency of a D2D communication link subject to four different combinations of transmitting and interference power constraints. To solve the original formulated nonlinear fractional programming problem, we first convert it into a dual nonlinear parametric programming problem, and then decouple the dual problem into several solvable concave problems. Further, a closed-form solution is derived to each dual problem and an efficient power allocation algorithm is proposed to find a numerical solution to the formulated problem. Simulation results show that the proposed power allocation algorithm outperforms an ergodic capacity maximization algorithm and a uniform power distribution algorithm in terms of the energy efficiency of a D2D link, and can efficiently improve the overall ergodic capacity of a cellular network.

SDO and LDO relaxation approaches to complex fractional quadratic optimization

This paper examines a complex fractional quadratic optimization problem subject to two quadratic constraints. The original problem is transformed into a parametric quadratic programming problem by the well-known classical Dinkelbach method. Then a semidefinite and Lagrangian dual optimization approaches are presented to solve the nonconvex parametric problem at each iteration of the bisection and generalized Newton algorithms. Finally, the numerical results demonstrate the effectiveness of the proposed approaches.