Kuhn-Tucker Research Articles

OPTIMIZATION OF A FUZZY INVENTORY MODEL WITH PENTAGONAL FUZZY NUMBERS

This paper explores an optimal replenishment strategy for a two-echelon inventory model which has been considered and analyzed in a fuzzy environment. In fuzzy environment, carrying cost, ordering cost and the replenishment processing cost are assumed to be pentagonal fuzzy numbers. The purpose of this model is to minimize the total inventory cost in fuzzy scenario. There are two inventory models proposed in this paper. Crisp models are developed with fuzzy total inventory cost but crisp optimal order quantity. Fuzzy model is also formulated with fuzzy total inventory cost and fuzzy optimal order quantity. Graded mean integration formula is employed to defuzzify the total inventory cost and the Kuhn–tucker condition is used to determine the optimal order quantity. Finally, we develop an algorithm to obtain the optimal order quantity. A comparison of fuzzy model with classical inventory model is being made. Numerical results highlighting the sensitivity of various parameters are also elucidated. The result illustrates that this fuzzy model can be quite useful in determining the optimal order quantity and minimum total inventory cost.

LU-Optimality Conditions in Optimization Problems With Mechanical Work Objective Functionals.

In this article, we introduce interval-valued Kuhn-Tucker (KT)-pseudoinvex optimization problems governed by interval-valued path-independent curvilinear integral objective functionals. Concretely, it is proven that an interval-valued KT-pseudoinvex variational control problem is described such that every KT point is an LU-optimal solution. In addition, the main results are highlighted by two illustrative applications describing the controlled behavior of an artificial neural system.

Optimization of Data Distributed Network System under Uncertainty

The major network design or data distributed problems may be described as constrained optimization problems. Constrained optimization problems include restrictions imposed by the system designers. These limitations are basically due to the system design’s physical limitations or functional requirements of the network system. Constrained optimization is a computationally challenging job whenever the constraints/limitations are nonlinear and nonconvex. Furthermore, nonlinear programming methods can easily deal same optimization problem if somehow the constraints are nonlinear and convex. In this paper, we have addressed a distributed network design problem involving uncertainty that transmits data across a parallel router. This distributed network design problem is a Jackson open‐type network design problem that has been formulated based on the M/M/1 queueing system. Because our network design problem is a nonlinear, convex optimization problem, we have employed a well‐known Kuhn–Tucker (K‐T) optimality algorithm to solve the same. Here, we have used triangular fuzzy numbers to express uncertain traffic rates and data processing rates. Then, by applying α‐level interval of fuzzy numbers and their corresponding parametric representation of α‐level intervals, the associated network design problem has been transformed to its parametric form and later has been solved. To obtain the optimal data stream rate in terms of interval and to illustrate the applicability of the entire approach, a hypothetical numerical example has been exhibited. Finally, the most important results have been reported.

Isolated efficiency in nonsmooth semi-infinite multi-objective programming

ABSTRACTIn this paper, isolated efficient solutions of a given nonsmooth Multi-Objective Semi-Infinite Programming problem (MOSIP) are studied. Two new Data Qualifications (DQs) are introduced and it is shown that these DQs are, to a large extent, weaker than already known Constraint Qualifications (CQs). The relationships between isolated efficiency and some relevant notions existing in the literature, including robustness, are established. Various necessary and sufficient conditions for characterizing isolated efficient solutions of a general problem are derived. It is done invoking the tangent cones, the normal cones, the generalized directional derivatives, and some gap functions. Using these characterizations, the (strongly) perturbed Karush-Kuhn-Tucker (KKT) optimality conditions for MOSIP are analyzed. Furthermore, it is shown that each isolated efficient solution is a Geoffrion properly efficient solution under appropriate assumptions. Moreover, Kuhn-Tucker (KT) and Klinger properly efficient solutions for a nonsmooth MOSIP are defined and it is proved that each isolated efficient solution is a KT properly efficient solution in general, and a Klinger properly efficient solution under a DQ. Finally, in the last section, the largest isolated efficiency constant for a given isolated efficient solution is determined.

Joint power and beam allocation of opportunistic array radar for multiple target tracking in clutter

This paper proposes a joint power and beam allocation scheme based on chance-constraint programming (CCP) for multiple target tracking of opportunistic array radar (OAR) in cluttered domains. The objective of the scheme is to cope with complex environments, unknown target information, and insufficient resource of radar system installed in mobile platforms operating over prolonged time periods. In this scheme, the total power of transmitting beams used in estimation process is minimized by effective allocation over the targets so that the desired tracking performance is attained. Allowing for the randomness of target RCS, the CCP used for balancing tracking performance and power resource is introduced conditioned on a specified confidence level. To track the targets more efficiently, the beams are allocated to the targets with large prior Cramér–Rao lower bound (CRLB). Then, in conjunction with an information reduction factor (IRF), the tracking Bayesian CRLB (BCRLB) of the illuminated targets can be considered as a measurement criteria for resource allocation in clutter. Through Conditional Value at Risk (CVaR), the CCP is relaxed to a convex problem solved by Lagrange multipliers method with Kuhn–Tucker (KT). The simulation results verify the validity and practicability of the resource allocation scheme.

KT‐pseudoinvex multidimensional control problem

SummaryIn this paper, we define a Kuhn‐Tucker (KT)–pseudoinvex multidimensional control problem. More exactly, we introduce a new condition on the functions, which are involved in a multidimensional control problem, and we prove that a KT‐pseudoinvex multidimensional control problem is characterized such that a KT point is an optimal solution. Thus, we generalize optimality results in known mathematical programming problems. These theoretical results are illustrated with an application.

Strictly feasible solutions and strict complementarity in multiple objective linear optimization

Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary conditions exist. Furthermore, we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary conditions for Isermann’s dual problem. Also, we give an alternative proof to establish necessary conditions for weakly efficient solutions of multiple objective programs, assuming the Kuhn–Tucker (KT) constraint qualification. We also provide a new condition to ensure the KT constraint qualification.

An Optimization Technique Based on Profit of Investment and Market Clearing in Wind Power Systems

Recently, renewable energies are widely used instead of the fuel energies due to their individual potentials such as its availability, low price and environmentally friendly. One of the most important renewable energies is wind power. As a result, investment in wind power is one of the most interesting research to maximize the profit of the investment and market clearing. In this paper, bi-level optimization technique is proposed to maximize the investment problem and market clearing for the wind power at the same time and in one single problem. Then, karush–kuhn–tucker (KKT) conditions and mathematical programming with equilibrium constraints (MPEC) are applied and tried to find one level optimization problem. Due to the nonlinearity of the optimization equation, the Fortuny-Amat & McCarl (FM) linearization technique is used to linearize the model. Finally, the proposed technique is applied to the IEEE 24 buses. The result proves that the optimization analysis is very easy, fast and accurate due to the linear characteristic of the system. All the simulation results are carried out in MATLAB and GAMS softwares.

The Research on Ticket Fare Optimization for China’s High-Speed Train

With the constant deepening of the railway reform in China, the competition between railway passenger transportation and other modes of passenger transportation has been getting fiercer. In this situation, the existing unitary and changeless fare structure gradually becomes the prevention of railway revenue increase and railway system development. This paper examines a new method for ticket fare optimization strategy based on the revenue management theory. On the premise that only one fare grade can be offered for each OD at the same time, this paper addresses the questions of how to determine the number of fare grades and the price of each fare grade. First, on the basis of piecewise pricing strategy, we build a ticket fare optimization model. After transforming this model to a convex program, we solve the original problem by finding the Kuhn-Tucker (K-T) point of the convex program. Finally, we verify the proposed method by real data of Beijing-Shanghai HSR line. The calculating result shows that our pricing strategy can not only increase the revenue, but also play a part in regulating the existing demand and stimulating potential demand.

Valuing hypothetical wildfire impacts with a Kuhn–Tucker model of recreation demand

This study uses a nonmarket valuation method to investigate the recreation values of the San Jacinto Wilderness in southern California. The analysis utilizes survey data from a stated-choice experiment involving backcountry visitors who responded to questions about hypothetical wildfire burn scenarios. Benefits of landscape preservation are derived using a Kuhn–Tucker (KT) demand system. Model results suggest that recreationists are attracted to sites with recent wildfires that can be viewed up-close. For example, recreational welfare estimates increased for sites that were partially affected by different types of wildfires, with the greatest gains being observed for the most recent wildfires. Per person mean seasonal willingness-to-pay varied from a low of $10 to a high of $48, for total gains ranging from $62,223 to $635,286. However, wildfires that cause trail closures create welfare losses. Seasonal losses per person for complete closure of particular sites range from $3 to $221, for total losses ranging from $29,600 to $2.9 million.

Dynamic user equilibrium based on a hydrodynamic model

In this paper we present a continuous-time network loading procedure based on the Lighthill–Whitham–Richards model proposed by Lighthill and Whitham (1955) and Richards (1956). A system of differential algebraic equations (DAEs) is proposed for describing traffic flow propagation, travel delay and route choices. We employ a novel numerical apparatus to reformulate the scalar conservation law as a flow-based partial differential equation (PDE), which is then solved semi-analytically with the Lax–Hopf formula. This approach allows for an efficient computational scheme for large-scale networks. We embed this network loading procedure into the dynamic user equilibrium (DUE) model proposed by Friesz et al. (1993). The DUE model is solved as a differential variational inequality (DVI) using a fixed-point algorithm. Several numerical examples of DUE on networks of varying sizes are presented, including the Sioux Falls network with a significant number of paths and origin–destination pairs (OD).The DUE model presented in this article can be formulated as a variational inequality (VI) as reported in Friesz et al. (1993). We will present the Kuhn–Tucker (KT) conditions for that VI, which is a linear system for any given feasible solution, and use them to check whether a DUE solution has been attained. In order to solve for the KT multiplier we present a decomposition of the linear system that allows efficient computation of the dual variables. The numerical solutions of DUE obtained from fixed-point iterations will be tested against the KT conditions and validated as legitimate solutions.

A hybrid PSO approach for solving non-convex optimization problems

A hybrid PSO approach for solving non-convex optimization problemsThe aim of this paper is to propose an improved particle swarm optimization (PSO) procedure for non-convex optimization problems. This approach embeds classical methods which are the Kuhn-Tucker (KT) conditions and the Hessian matrix into the fitness function. This generates a semi-classical PSO algorithm (SPSO). The classical component improves the PSO method in terms of its capacity to search for optimal solutions in non-convex scenarios. In this work, the development and the testing of the refined the SPSO algorithm was carried out. The SPSO algorithm was tested against two engineering design problems which were; ‘optimization of the design of a pressure vessel’ (P1) and the ‘optimization of the design of a tension/compression spring’ (P2). The computational performance of the SPSO algorithm was then compared against the modified particle swarm optimization (PSO) algorithm of previous work on the same engineering problems. Comparative studies and analysis were then carried out based on the optimized results. It was observed that the SPSO provides a better minimum with a higher quality constraint satisfaction as compared to the PSO approach in the previous work.

KKT-based iterative algorithm for beamforming in cognitive radio system

In this paper, we consider the downlink channel of multi-user multi-input single-output (MU-MISO) system in cognitive radio network, where the cognitive base station (CBS) resort to beamforming scheme to relief co-channel interference. The design criterion is to minimize the transmit power at CBS, subject to the signal-to-interference-plus-noise-ratio (SINR) constraints of cognitive users (CUs) and the interference constraints at primary users (PUs). Standard conic optimization packages can handle the problem, however, the complexity is very high and optimization packages are not always available. Basing on the karush–kuhn–tucker (KKT) conditions of the converted optimization problem, we proposed an iteration algorithm. Simulation results reveal that the proposed algorithm can converge to the optimal beamforming vectors that lead to minimum transmit power with all constraints satisfied.

A latent segmentation approach to a Kuhn–Tucker model: An application to recreation demand

In this paper, we extend the latent segmentation approach to the Kuhn–Tucker (KT) model. The proposed approach models heterogeneity in preferences for recreational behavior, using a utility theoretical framework to simultaneously model participation and site selection decisions. Estimation of the latent segmentation KT model with standard maximum likelihood techniques is numerically difficult because of the large number of parameters in the segment membership functions and the utility function for each latent segment. To address this problem, we propose the expectation-maximization (EM) algorithm to estimate the model. In the empirical section, we implement the EM latent segmentation KT approach to analyze a Southern California beach recreation data set. Our empirical analysis suggests that three groups exist in the sample. Using the model to analyze two hypothetical beach management policy scenarios illustrates different welfare impacts across groups.

Mean-variance models for portfolio selection with fuzzy random returns

This paper develops two novel types of mean-variance models for portfolio selection problems, in which the security returns are assumed to be characterized by fuzzy random variables with known possibility and probability distributions. In the proposed models, we take the expected return of a portfolio as the investment return and the variance of the expected return of a portfolio as the investment risk. We assume that the security returns are triangular fuzzy random variables. To solve the proposed portfolio problems, this paper first presents the variance formulas for triangular fuzzy random variables. Then this paper applies the variance formulas to the proposed models so that the original portfolio problems can be reduced to nonlinear programming ones. Due to the reduced programming problems include standard normal distribution in the objective functions, we cannot employ the conventional solution methods to solve them. To overcome this difficulty, this paper employs genetic algorithm (GA) to solve them, and verify the obtained optimal solutions via Kuhn-Tucker (K-T) conditions. Finally, two numerical examples are presented to demonstrate the effectiveness of the proposed models and methods.

The multiple discrete-continuous extreme value (MDCEV) model: Role of utility function parameters, identification considerations, and model extensions

Many consumer choice situations are characterized by the simultaneous demand for multiple alternatives that are imperfect substitutes for one another. A simple and parsimonious multiple discrete-continuous extreme value (MDCEV) econometric approach to handle such multiple discreteness was formulated by Bhat (2005) [Bhat, C.R., 2005. A multiple discrete-continuous extreme value model: formulation and application to discretionary time-use decisions. Transportation Research Part B 39(8), 679–707]. within the broader Kuhn–Tucker (KT) multiple discrete-continuous economic consumer demand model of Wales and Woodland (1983) [Wales, T.J., and Woodland, A.D., 1983. Estimation of consumer demand systems with binding non-negativity constraints. Journal of Econometrics 21(3), 263–85]. This paper examines several issues associated with the MDCEV model and other extant KT multiple discrete-continuous models. Specifically, the paper proposes a new utility function form that enables clarity in the role of each parameter in the utility specification, presents identification considerations associated with both the utility functional form as well as the stochastic nature of the utility specification, extends the MDCEV model to the case of price variation across goods and to general error covariance structures, discusses the relationship between earlier KT-based multiple discrete-continuous models, and illustrates the many technical nuances and identification considerations of the multiple discrete-continuous model structure through empirical examples. The paper also highlights the technical problems associated with the stochastic specification used in the KT-based multiple discrete-continuous models formulated in recent Environmental Economics papers.

Normal vector identification and interactive tradeoff analysis using minimax formulation in multiobjective optimization

In multiobjective optimization, tradeoff analysis plays an important role in determining the best search direction to reach a most preferred solution. This paper presents a new explicit interactive tradeoff analysis method based on the identification of normal vectors on a noninferior frontier. The interactive process is implemented using a weighted minimax formulation by regulating the relative weights of objectives in a systematic manner. It is proved under a mild condition that a normal vector can be identified using the weights and Kuhn-Tucker (K-T) multipliers in the minimax formulation. Utility gradients can be estimated using local preference information such as marginal rates of substitution. The projection of a utility gradient onto a tangent plane of the noninferior frontier provides a descent direction of disutility and thereby a desirable tradeoff direction, along which tradeoff step sizes can be decided by the decision maker using an explicit tradeoff table. Necessary optimality conditions are established in terms of normal vectors and utility gradients, which can be used to guide the elicitation of local preferences and also to terminate an interactive process in a rigorous yet flexible way. This method is applicable to both linear and nonlinear (either convex or nonconvex) multiobjective optimization problems. Numerical examples are provided to illustrate the theoretical results of the paper and the implementation of the proposed interactive decision analysis process.

New Conditions for Exactness of a Simple Penalty Function

We consider penalty function methods for finding the maximum of a function f over the set \[ S = \{ x \in R^n :g_i (x) \leqq 0{\text{ for }}i = 1, \cdots ,m{\text{ and }}h_j (x) = 0{\text{ for }}j = 1, \cdots ,p\} . \] New conditions, extending earlier work done by Pietrzykowski, are presented under which the penalty function \[ P(x,\mu ) = \mu f(x) - \sum\limits_{i - 1}^m {g_i^ + } (x) - \sum\limits_{j = 1}^p {\left| {h_j (x)} \right|} \] is locally exact. The relationships among the new conditions, Pietrzykowski’s conditions, and Kuhn–tucker constraint qualifications are explored.

Limitations and possibilities for improvements of photovoltaic solar energy converters : Wolf, M. The Proceedings of the IRE, July 1960, p. 20. illus.

This paper proposes a joint power and beam allocation scheme based on chance-constraint programming (CCP) for multiple target tracking of opportunistic array radar (OAR) in cluttered domains. The objective of the scheme is to cope with complex environments, unknown target information, and insufficient resource of radar system installed in mobile platforms operating over prolonged time periods. In this scheme, the total power of transmitting beams used in estimation process is minimized by effective allocation over the targets so that the desired tracking performance is attained. Allowing for the randomness of target RCS, the CCP used for balancing tracking performance and power resource is introduced conditioned on a specified confidence level. To track the targets more efficiently, the beams are allocated to the targets with large prior Cramér–Rao lower bound (CRLB). Then, in conjunction with an information reduction factor (IRF), the tracking Bayesian CRLB (BCRLB) of the illuminated targets can be considered as a measurement criteria for resource allocation in clutter. Through Conditional Value at Risk (CVaR), the CCP is relaxed to a convex problem solved by Lagrange multipliers method with Kuhn–Tucker (KT). The simulation results verify the validity and practicability of the resource allocation scheme.