Kuhn Tucker Optimality Conditions Research Articles

Optimality Conditions and Duality of Three Kinds of Nonlinear Fractional Programming Problems

Some assumptions for the objective functions and constraint functions are given under the conditions of convex and generalized convex, which are based on theF-convex,ρ-convex, and(F,ρ)-convex. The sufficiency of Kuhn-Tucker optimality conditions and appropriate duality results are proved involving(F,ρ)-convex,(F,α,ρ,d)-convex, and generalized(F,α,ρ,d)-convex functions.

The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function

In this paper, we study the Karush–Kuhn–Tucker optimality conditions in a class of nonconvex optimization problems with an interval-valued objective function. Firstly, the concepts of preinvexity and invexity are extended to interval-valued functions. Secondly, several properties of interval-valued preinvex and invex functions are investigated. Thirdly, the KKT optimality conditions are derived for LU-preinvex and invex optimization problems with an interval-valued objective function under the conditions of weakly continuous differentiablity and Hukuhara differentiablity. Finally, the relationships between a class of variational-like inequalities and the interval-valued optimization problems are established.

Podmienky optimálnosti Kuhna-Tuckera v modeloch rovnováhy trhu sieťových odvetví

Currently a considerable attention to the subject of network industries is being paid in discussions on various levels. It is understandable as network industries in fact ensure the production and distribution of energy sources which play a key role in developed economies. The discussions are usually focused on the question of reasonable profit of network industries subjects and on the other hand the question of generally acceptable costs. Equilibrium on the network industries market, as well as on any market, is being created based on the level of demand and supply on relevant market. In this article we will discuss the analysis of optimization models of consumers and producers behavior on the network industries market as well as the question of effectiveness of this specific market. We will point out certain features of network industries market where the consumer usually is not able to substitute a product of network industry with other product of appropriate characteristics in a short time period, thus considering the product being exclusive. This exclusivity can be formally represented in the utility function and other related analytical tasks. In paper we study the properties of a network industry optimization problem and economically interpretable implications of Kuhn-Tucker optimality conditions of this model.

Optimality conditions in a vector continuous-time optimization problem

We consider nonlinear vector continuous-time optimization problems with inequality type of constraints. We derive new Fritz John and Karush–Kuhn–Tucker optimality conditions that are valid when all functions are preinvex.

Optimal design of rectangular RC sections for ultimate bending strength

A minimum cost problem for ultimate strength in bending of rectangular reinforced concrete sections is investigated. The design variables are section depth and steel reinforcement areas. State equations are those of equilibrium with compression depth as state variable. The Kuhn-Tucker optimality conditions are solved analytically and formulas for nondimensional design and state variables are obtained in four cases: Two singly-reinforced solutions with either maximum allowable depth or smaller; Two doubly-reinforced with maximum allowable depth and either maximum compression depth or smaller. Each of the solutions is optimal in a region of the plane `nondimensional bending moment'---`cost-effectiveness ratio of concrete to steel'. The formulas are for an arbitrary concrete constitutive law with tension cut-off and are specialized for the parabola-rectangle law of Eurocode 2.

The Karush–Kuhn–Tucker optimality conditions in minimum weight design of elastic rotating disks with variable thickness and density

The Karush–Kuhn–Tucker optimality conditions in minimum weight design of elastic rotating disks with variable thickness and density

An approach for the reliability based design optimization of laminated composites

This article aims at optimizing laminated composite plates taking into account uncertainties in the structural dimensions. As laminated composites require a global optimization tool, the Particle Swarm Optimization (PSO) method is employed. A new Reliability Based Design Optimization (RBDO) methodology based on safety factors is presented and coupled with PSO. Such safety factors are derived from the Karush–Kuhn–Tucker optimality conditions of the reliability index approach and eliminate the need for reliability analysis in RBDO. The plate weight minimization is the objective function of the optimization process. The results show that the coupling of the evolutionary algorithm with the safety-factor method proposed in this article successfully performs the RBDO of laminated composite structures.

An incremental primal–dual method for nonlinear programming with special structure

We propose a new class of incremental primal–dual techniques for solving nonlinear programming problems with special structure. Specifically, the objective functions of the problems are sums of independent nonconvex continuously differentiable terms minimized subject to a set of nonlinear constraints for each term. The technique performs successive primal–dual increments for each decomposition term of the objective function. The primal–dual increments are calculated by performing one Newton step towards the solution of the Karush–Kuhn–Tucker optimality conditions of each subproblem associated with each objective function term. We show that the resulting incremental algorithm is q-linearly convergent under mild assumptions for the original problem.

Utility-directed resource allocation in virtual desktop clouds

User communities are rapidly transitioning their “traditional desktops” that have dedicated hardware and software installations into “virtual desktop clouds” (VDCs) that are accessible via thin-clients. To allocate and manage VDC resources for Internet-scale desktop delivery, existing works focus mainly on managing server-side resources based on utility functions of CPU and memory loads, and do not consider network health and thin-client user experience. Resource allocations without combined utility-directed information of system loads, network health and thin-client user experience in VDC platforms inevitably results in costly guesswork and over-provisioning of resources. In this paper, we develop an analytical model viz., “Utility-Directed Resource Allocation Model (U-RAM)” to solve the combined utility-directed resource allocation problem within VDCs. Our solution involves an iterative algorithm that leverages utility functions of system, network and human components obtained using a novel virtual desktop performance benchmarking toolkit viz., “VDBench” that we developed. The combined utility functions are used to direct decision schemes based on Kuhn–Tucker optimality conditions for creating user desktop pools and determining optimal resource allocation size/location. We deploy VDBench in a VDC testbed featuring: (a) popular user applications (Spreadsheet Calculator, Internet Browser, Media Player, Interactive Visualization), and (b) TCP/UDP based thin-client protocols (RDP, RGS, PCoIP) under a variety of user load and network health conditions. Simulation results based on the utility functions obtained from the testbed demonstrate that our solution maximizes VDC scalability i.e., ‘VDs per core density’, and ‘user connections quantity’, while delivering satisfactory thin-client user experience.

Cycle-breaking acceleration for support vector regression

Support vector regression (SVR) is a powerful tool in modeling and prediction tasks with widespread application in many areas. The most representative algorithms to train SVR models are Shevade et al.'s Modification 2 and Lin's WSS1 and WSS2 methods in the LIBSVM library. Both are variants of standard SMO in which the updating pairs selected are those that most violate the Karush–Kuhn–Tucker optimality conditions, to which LIBSVM adds a heuristic to improve the decrease in the objective function. In this paper, and after presenting a simple derivation of the updating procedure based on a greedy maximization of the gain in the objective function, we show how cycle-breaking techniques that accelerate the convergence of support vector machines (SVM) in classification can also be applied under this framework, resulting in significantly improved training times for SVR.

Study on Multi-Void Microstructure of Asymptotic Homogenization for Topology Optimization

One kind of multi-void three-dimensional microstructure models based on homogenization method is constructed. Based on multi-void microstructure, the mathematical models for the topological structural optimization which takes maximizing the total potential energy as the objective function is constructed, then the Kuhn-Tucker optimality condition of the update method about the designs variable based on the porous microstructure can be gotten when optimization iterates. Finally, in explaining the employed algorithm an example is provided.

Solving quadratic convex bilevel programming problems using a smoothing method

In this paper, we present a smoothing sequential quadratic programming to compute a solution of a quadratic convex bilevel programming problem. We use the Karush–Kuhn–Tucker optimality conditions of the lower level problem to obtain a nonsmooth optimization problem known to be a mathematical program with equilibrium constraints; the complementary conditions of the lower level problem are then appended to the upper level objective function with a classical penalty. These complementarity conditions are not relaxed from the constraints and they are reformulated as a system of smooth equations by mean of semismooth equations using Fisher–Burmeister functional. Then, using a quadratic sequential programming method, we solve a series of smooth, regular problems that progressively approximate the nonsmooth problem. Some preliminary computational results are reported, showing that our approach is efficient.

Exact penalty method for the nonlinear bilevel programming problem

In this paper, following the method of replacing the lower level problem with its Kuhn-Tucker optimality condition, we transform the nonlinear bilevel programming problem into a normal nonlinear programming problem with the complementary slackness constraint condition. Then, we get the penalized problem of the normal nonlinear programming problem by appending the complementary slackness condition to the upper level objective with a penalty. We prove that this penalty function is exact and the penalized problem and the nonlinear bilevel programming problem have the same global optimal solution set. Finally, we propose an algorithm for the nonlinear bilevel programming problem. The numerical results show that the algorithm is feasible and efficient.

On the optimality of some classes of invex problems

In this paper we define two notions: Kuhn–Tucker saddle point invex problem with inequality constraints and Mond–Weir weak duality invex one. We prove that a problem is Kuhn–Tucker saddle point invex if and only if every point, which satisfies Kuhn–Tucker optimality conditions forms together with the respective Lagrange multiplier a saddle point of the Lagrange function. We prove that a problem is Mond–Weir weak duality invex if and only if weak duality holds between the problem and its Mond–Weir dual one. Additionally, we obtain necessary and sufficient conditions, which ensure that strong duality holds between the problem with inequality constraints and its Wolfe dual. Connections with previously defined invexity notions are discussed.

A computationally efficient optimisation-based method for parameter identification of kinematically determinate and over-determinate biomechanical systems

This paper introduces a general optimisation-based method for identification of biomechanically relevant parameters in kinematically determinate and over-determinate systems from a given motion. The method is designed to find a set of parameters that is constant over the time frame of interest as well as the time-varying system coordinates, and it is particularly relevant for biomechanical motion analysis where the system parameters can be difficult to accurately determine by direct measurements. Although the parameter identification problem results in a large-scale optimisation problem, we show that, due to a special structure in the linearised Karush–Kuhn–Tucker optimality conditions, the solution can be found very efficiently. The method is applied to a set of test problems relevant for gait analysis. These involve determining the local coordinates of markers placed on the model, segment lengths and joint axes of rotation from both gait and range of motion experiments.

Infer objective function of glycerol metabolism in klebsiella pneumoniae basing on bilevel programming

Mathematical modelling of cellular metabolism plays an important role in understanding biological functions and providing identification of targets for biotechnological modification. This paper proposes a nonlinear bilevel programming (NBP) model to infer the objective function of anaerobic glycerol metabolism in Klebsiella Pneumoniae (K. Pneumoniae) for 1, 3-propanediol (1, 3-PD) production. Based on the Kuhn-Tucker optimality condition of the lower level problem, NBP is transformed into a nonlinear programming with complementary and slackness conditions. The authors give the existence theorem of solutions to NBP. An efficient algorithm is proposed to solve NBP and its convergence is also simply analyzed. Numerical results reveal some interesting conclusions, e.g., biomass production is the main force to drive glycerol metabolism, and the objective functions, which are obtained in term of several different groups of flux distributions, are similar.

Bilevel programming applied to power system vulnerability analysis under multiple contingencies

This study examines the use of bilevel programming to analyse the vulnerability of power systems under multiple contingencies. One of the main purposes of this study is to explain the state of the art of the subject matter. A minimum vulnerability model and a maximum vulnerability model are presented and discussed. In both models, the upper-level optimisation determines a set of simultaneous outages in the transmission network whereas the lower-level optimisation models the reaction of the system operator against the outages identified in the upper level. The system operator reacts by minimising the system load shed through an optimal operation of the power system. Two solution approaches for the resulting mixed-integer non-linear bilevel programs are analysed and compared. Both methodologies are based on the equivalent transformation of the lower-level problem into a set of constraints, so that the original bilevel programs, respectively, become a single-level optimisation problem. The first approach is based on the application of Karush–Kuhn–Tucker optimality conditions whereas the second procedure relies on duality theory. This study shows that both approaches are essentially equivalent from a rigorous mathematical viewpoint; however, the second method is more suitable for off-the-shell branch-and-cut software as corroborated by numerical simulations.

Probabilistic Analysis of Semidefinite Relaxation for Binary Quadratic Minimization

We consider semidefinite programming relaxation (SDR) of a binary quadratic minimization problem. This NP-hard problem arises naturally in the maximum-likelihood detection of discrete signals for digital communications. We analyze the average performance of the SDR algorithm for a class of randomly generated binary quadratic minimization problems. Although the SDR worst-case approximation ratio is unbounded for this NP-hard problem, our analysis shows that SDR can provide in polynomial time a provably near-optimal solution, achieving a constant factor approximation of the optimal objective value in probability. Moreover, this constant factor remains bounded with increasing problem size. Our proof is based on an asymptotic analysis of Karush–Kuhn–Tucker optimality conditions using random matrix theory.

Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems

By using the concept of Frechet differentiability of mapping, we present the Kuhn-Tucker optimality conditions for weakly efficient solution, Henig efficient solution, superefficient solution, and globally efficient solution to the vector equilibrium problems with constraints.

Kinematic analysis of over-determinate biomechanical systems

In this paper, we introduce a new general method for kinematic analysis of rigid multi body systems subject to holonomic constraints. The method extends the standard analysis of kinematically determinate rigid multi body systems to the over-determinate case. This is accomplished by introducing a constrained optimisation problem with the objective function given as a function of the set of system equations that are allowed to be violated while the remaining equations define the feasible set. We show that exact velocity and acceleration analysis can also be performed by solving linear sets of equations, originating from differentiation of the Karush–Kuhn–Tucker optimality conditions. The method is applied to the analysis of an 18 degrees-of-freedom gait model where the kinematical drivers are prescribed with data from a motion capture experiment. The results show that significant differences are obtained between applying standard kinematic analysis or minimising the least-square errors on the two fully equivalent 3D gait models with only the way the experimental data is processed being different.