Parametric Programming Problem Research Articles

On solving quadratically constrained quadratic programming problem with one non-convex constraint

In this paper we consider a quadratically constrained quadratic programming problem with convex objective function and many constraints in which only one of them is non-convex. This problem is transformed to a parametric quadratic programming problem without any non-convex constraint and then by solving the parametric problem via an iterative scheme and updating the parameter in each iteration, the solution of the problem is achieved. The convergence of the proposed method is investigated. Numerical examples are given to show the applicability of the new method.

Optimizing a Fuzzy Green p-hub Centre Problem Using Opposition Biogeography Based Optimization

Hub networks have always been acriticalissue in locating health facilities. Recently, a study has been investigated by Cocking et al. (2006)in Nouna health district in Burkina Faso, Africa, with a population of approximately 275,000 people living in 290 villages served by 23 health facilities. The travel times of the population to health services become extremely high during the rainy season, since many roads are unusable. In this regard, for many people, travelling to a health facility is a deterrent to seeking proper medical care. Furthermore, in real applications of hub networks, the travel times may vary due to traffic, climate conditions, and land or road type.To handle this challenge this paper considers the travel times are assumed to be characterized by trapezoidal fuzzy variables in order to present a fuzzy green capacitated single allocation p-hub center system (FGCSApHCP) with uncertain information. The proposed FGCSApHCP is redefined into its equivalent parametric integer nonlinear programming problem using credibility constraints. The aim is to determine the location of pcapacitated hubs and the allocation of center nodes to them in order to minimize the maximum travel time in a hub-and-center network in such uncertain environment. As the FGCSApHCP is NP-hard, a novel algorithmcalledoppositionbiogeography based optimizationis developed to solve that. This algorithm utilizes a binary oppositionbased learning mechanism to generate a diversity mechanism. At the end, both the applicability of the proposed approach and the solution methodologies are demonstrated using GAMS/BARON Software under severalkind of problems. Sensitivity analyses on the number of hubs and center nodes are conducted toprovide more insights as well.

Natural-Gas-Derived Hydrogen in the Presence of Carbon Fuel Taxes and Concentrated Solar Power

This work focuses on the incorporation of renewable energy resources into the natural-gas-based production of hydrogen, via steam methane reforming (SMR). The novel concept of “energetically enhanced steam methane reforming” (EESMR) is introduced, which changes the endothermicity level of the SMR process through incorporation of carbon monoxide and steam into the SMR feed. Novel EESMR flow sheets are presented in which the aforementioned resources are internally generated and recycled, so as to create a natural-gas-based hydrogen production system with methane as raw material and a hybrid (methane/solar) energy supply. The current worldwide carbon tax legislative environment is briefly reviewed, and its potential impact on SMR-versus-EESMR economic viability is quantified. A novel method for solution of linear parametric programming problems is proposed based on the concept of dimensionality reduction—this allows the analytic quantification of the optimum objective function value and associated optimum va...

Energy Efficiency Optimization for MIMO Distributed Antenna Systems With Pilot Contamination

In this paper, we study the energy-efficiency (EE) maximization problem for a multiple-input multiple-output distributed antenna system (DAS) with pilot contamination. With per-user quality of service constraints and per remote antenna unit (RAU) power requirements, we formulate the EE maximization problem as a joint optimization of sparse transmit beamforming, RAU selection, and RAU clustering. The considered problem is a non-convex multivariate optimization problem. To solve the problem, we transform it to an equivalent parametric programming problem (PPP) with a given EE parameter and design a two-layer optimization scheme to solve the original problem. The outer layer involves two kinds of algorithms to iteratively update the EE parameter based on Dinkelbach’s algorithm and bi-section search, respectively. The more challenging issue lies in the inner loop, where a non-convex multivariate PPP needs to be tackled. A series of techniques, including the reweighted $\ell_\mathbf {1}$ -norm, D.C. function, and semidefinite relaxation (SDR), is adopted to approximate the non-convex multivariate PPP with a convex SDR problem. Furthermore, a heuristic algorithm is proposed to reduce the complexity of a two-layer scheme. Simulation results show that the proposed algorithms significantly improve the EE and demonstrate that RAU selection and RAU clustering contribute to a higher EE.

Fully Decentralized Optimal Power Flow of Multi-Area Interconnected Power Systems Based on Distributed Interior Point Method

Traditionally, the optimal power flow (OPF) problem is solved in a centralized manner. However, with continuous expansion of the scale of multi-area interconnected power systems, realistic applications of the centralized method face additional challenges. In this paper, we propose a fully decentralized OPF algorithm for multi-area interconnected power systems based on the distributed interior point method, where solving the regional correction equation was converted into solving a parametric quadratic programming problem during each Newton-Raphson iteration. This is a novel approach to tackle the decentralized OPF problems. We analyzed the convergence property of this algorithm. The proposed approach is fully decentralized without central coordinator and parameter tuning, and is robust to network partitioning. Furthermore, this decentralized algorithm enjoys the same convergence performance and accuracy as the centralized interior point method. Results on a 3-bus test system, 4 IEEE test systems, and a real 4-area 6056-bus interconnected system show the benefits of the proposed method.

Autonomous Transmission Power Decision Strategy for Energy Efficient Operation of a Dense Small Cell Network

Smart interference management methods are required to enhance the throughput, coverage, and energy efficiency of a dense small cell network. In this paper, we propose a transmit power control for energy efficient operation of a dense small cell network. We cast the power control problem as a noncooperative game to satisfy the design requirement that small cells do not need any information exchange among them. We analyze the sufficient condition for the existence of a Nash equilibrium (NE) state of the proposed game. We also analyze that the NE state is unique by transforming the original nonlinear fractional programming problem into a nonlinear parametric programming problem. Through simulation studies, we verify our analysis results. In addition, we show that the proposed method achieves higher energy efficiency of a network and balances the energy efficiency among cells more evenly than the methods based on the AIMD (additive increase and multiplicative decrease) algorithm.

A Solution Path Algorithm for General Parametric Quadratic Programming Problem.

Parameter in learning problems (usually arising from the tradeoff between training error minimization and regularization) is often tuned by cross validation (CV). A solution path provides a compact representation of all optimal solutions, which can be used to determine the parameter with the global minimum CV error, without solving original optimization problems multiple times based on grid search. However, existing solution path algorithms do not provide a unified implementation to various learning problems. In this paper, we first introduce a general parametric quadratic programming (PQP) problem that can be instantiated to an extensive number of learning problems. Then, we propose a generalized solution path (GSP) for the general PQP problem. Particularly, we use the $QR$ decomposition to handle singularities in GSP. Finally, we analyze the finite convergence and the time complexity of GSP. Our experimental results on a variety of data sets not only confirm the identicality between GSP and several existing solution path algorithms but also show the superiority of our GSP over the existing solution path algorithms on both generalization and robustness. Finally, we provide a practical guild of using the GSP to solve two important learning problems, i.e., generalized error path and Ivanov SVM.

On multivariate quantile regression analysis

This paper investigates the estimation of parameters in a multivariate quantile regression model when the investigator wants to evaluate the associated distribution function. It proposes a new directional quantile estimator with the following properties: (1) it applies to an arbitrary number of random variables; (2) it is equivalent to estimating the distribution function allowing for non-convex distribution contours; (3) it satisfies nice equivariance properties; (4) it has desirable statistical properties (i.e., consistency and asymptotic normality); and (5) its implementation involves a modest computational burden: our proposed estimator can be obtained by solving parametric linear programming problems. As such, this paper expands the range of applications of quantile estimation for multivariate regression models.

Exact solution to a parametric linear programming problem

We consider the problem of determining the interval hull solution l ∗ to a parametric linear programming (PLP) problem l(x, p) = c T (p)x, where c i (p) can be in general nonlinear functions of p, and x satisfy the constraint A(p)x = b(p), p ∈ p. A new iterative method for determining l ∗ is suggested. The method exploits the concept of the p-solution to a square linear interval parametric (LIP) system, having the parametric form x(p) = L p + a, p ∈ p, where L is a real n × m matrix (n and m are, respectively, the sizes of the square matrix A and vector p), whereas a is an interval vector. A numerical example is provided to illustrate the new approach.

The Mordukhovich Coderivative and the Local Metric Regularity of the Solution Map to a Parametric Discrete Optimal Control Problem

In this paper, we study the Mordukhovich coderivative and the local metric regularity in Robinson’s sense of the solution map to a parametric dynamic programming problem with linear constraints and convex cost functions. By establishing abstract results on the coderivative and the local metric regularity of the solution map to a parametric variational inequality, we obtain the Mordukhovich coderivative and the local metric regularity in Robinson’s sense of the solution map to a parametric discrete optimal control problem.

An Efficient Algorithm for Quadratic Sum-of-Ratios Fractional Programs Problem

ABSTRACTThe quadratic sum-of-ratios fractional program problem has a broad range of applications in practical problems. This article will present an efficient branch-and-bound algorithm for globally solving the quadratic sum-of-ratios fractional program problem. In this algorithm, lower bounds are computed by solving a series of parametric relaxation linear programming problems, which are established by utilizing new parametric linearizing technique. To enhance the computational speed of the proposed algorithm, a rectangle reducing tactic is used to reject a part of the investigated rectangle or the whole rectangle where there does not contain any global optimal solution of the quadratic sum-of-ratios fractional program problem. Compared with the known approaches, the proposed algorithm does not need to introduce new variables and constraints. Therefore, the proposed algorithm is more suitable for application in engineering.

Energy states aided relay selection and optimal power allocation for cognitive relaying networks

Energy harvesting (EH) is a promising technique for cognitive relaying transmission (CRT) where secondary users (SUs) and relaying nodes do not have a constant power supply each. Unlike conventional CRT where the end-to-end data rate is usually maximised without taking into account the energy consumption at the source and relay, in this study, the energy consumption is characterised by means of energy efficiency, defined as the achievable data rate per Joule. In particular, the energy states at each node (either at a SU or a relay) is modelled as a finite-state Markov chain and the transmit power at a node is optimally allocated by jointly accounting for the interference threshold prescribed by primary users (PUs), the maximum allowable transmit power and the harvested energy at the node. To maximise the energy efficiency, a best relay selection criterion is proposed and the subsequent optimal transmit power allocation is initially formulated as a non-linear fractional programming problem and, then, equivalently transformed into a parametric programming problem and, finally, solved analytically by using the classic Karush–Kuhn–Tucker conditions. With extensive Monte-Carlo simulation results, the effectiveness of the proposed relay selection algorithm and corresponding optimal power allocation strategy are corroborated.

Hub-and-spoke network design problem under uncertainty considering financial and service issues: A two-phase approach

This paper proposes a bi-objective hub-and-spoke (H&S) network design problem with type-2 (T2) fuzzy transportation cost and travel time described by parametric secondary possibility distributions, which are obtained using three types of mean value (MV) reduction methods. The considered objectives jointly minimize the generalized expectation of the total transportation costs and the maximum travel time requirement in terms of generalized value-at-risk (VaR). To solve the fuzzy bi-objective H&S network design problem, we develop a two-phase approach, where in the first phase we convert the proposed model into its equivalent parametric mixed-integer programming problems by applying an equivalent transformation method. This is followed by the second phase using a fuzzy linear programming approach implemented with an augmented max-min operator to obtain a non-dominated solution that has an equal satisfactory degree on both objectives. Finally, a case study based on the Civil Aeronautics Board (CAB) data set is conducted to demonstrate the effectiveness of the proposed model and solution approach.

An Inner-Outer Approximation Approach to Chance Constrained Optimization

Nonlinear chance constrained optimization (CCOPT) problems are known to be difficult to solve. This work proposes a smooth approximation approach consisting of an inner and an outer analytic approximation of chance constraints. In this way, CCOPT is approximated by two parametric nonlinear programming (NLP) problems which can be readily solved by an NLP solver. Any optimal solution of the inner approximation problem is a priori feasible to the CCOPT. The solutions of the inner and outer problems, respectively, converge asymptotically to the optimal solution of the CCOPT.

On Fuzzy Portfolio Selection Problems: A Parametric Representation Approach

Fuzzy portfolio selection problem is a major issue in the financial field and a special case of constrained fuzzy-valued optimization problems (CFOPs). In this respect, the present paper aims to investigate the CFOP with regard to the features of the parametric representation of fuzzy numbers named as convex constraint function (CCF) which is proposed by Chalco-Cano et al. in 2014. Furthermore, relying on this parametric representation, some proper conditions are provided for the existence of solutions to a CFOP. To this end, by the increasing representation of CCF, the main problem is converted to a parametric multiobjective programming problem and some solution concepts from a similar framework in the multiobjective programming are proposed for the CFOP. Eventually to illustrate the proposed results, the fuzzy portfolio selection problem is discussed.

A study of one class of NLP problems arising in parametric semi-infinite programming

The paper deals with a nonlinear programming (NLP) problem that depends on a finite number of integers (parameters). This problem has a special form, and arises as an auxiliary problem in study of solutions' properties of parametric semi-infinite programming (SIP) problems with finitely representable compact index sets. Therefore, it is important to provide a deep study of this NLP problem and its properties w.r.t. the values of the parameters. We are especially interested in the case when optimal solutions of the NLP problem satisfy certain properties due to some specific requirements arising in parametric SIP. We establish the values of the parameters for which optimal solutions of the corresponding NLP problem fulfil the needed properties, and suggest an algorithm that determines the right values of the parameters. An example is proposed to illustrate the application of the algorithm.

Operational scheduling of microgrids via parametric programming

We present a parametric programming based approach for energy management in microgrids. An operational planning problem for a grid-connected microgrid with energy sources including solar photovoltaic, wind turbine and battery energy storage system, in addition to a household load demand, is captured as a parametric mixed-integer linear programming problem (p-MILP) through parameterizations of the uncertain coordinates of wind and solar energy resources. Thus, the energy management problem - typically nonlinear - is transformed into a linear bi-level optimization problem, where choice of the parameterization scheme is made at the upper level while system operation decisions are made at the lower level. The p-MILP formulation leads to significant improvements in uncertainty handling, solution quality and computational ease; by removing dependency of the solution on meteorological forecasts and avoiding the multiple computational cycles of the traditional online optimization techniques. The problem is solved offline on a flexible time-scale basis, allowing online implementation to be achievable on real-time system state updates. The proposed parametric programming approach extends the state-of-the-art in microgrid energy management methods and the results from various case studies are used to demonstrate the feasibility of our method.

On the achievement stability set for parametric linear goal programming problems

Goal Programming (GP) has many significant advances in recent years, particularly in the area of intelligent modeling and solution analysis as it is a powerful and effective methodology for the modeling, solution, and analysis of problems having multiple and conflicting objectives. This paper presents an algorithm for investigating the achievement stability set for parametric linear goal programming problems when parameters exist in the objective function. To avoid the complexity of this problem, the iterative approach in Dauer and Krueger (Oper Res Quart 28(3 Part 2):671–681, 1977) will be used to solve it. This model has been handled because it is very easy for the decision maker to put the weights for goals but, for the sub-goals it will be difficult as they have the same measure.

Subgradients of the Value Function in a Parametric Convex Optimal Control Problem

Motivated by our recent works on the optimal value function in parametric optimal control problems under linear state equations, in this paper we study of the first-order behavior of the value function of a parametric convex optimal control problem with a convex cost function and linear state equations. By establishing an abstract result on the subdifferential of the value function to a parametric convex mathematical programming problem, we derive a formula for computing the subdifferential and the singular subdifferential of the value function to a parametric convex optimal control problem. By virtue of the convexity, several assumptions used in the above papers, like the existence of a local upper Lipschitzian selection of the solution map, as well as the V-inner semicontinuity of the solution map, are no longer needed.

On Parametric Multi-level Multi-objective Fractional Programming Problems with Fuzziness in the Constraints

On Parametric Multi-level Multi-objective Fractional Programming Problems with Fuzziness in the Constraints