A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

Priority approach based on quadratic programming model to fuzzy preference relation

We investigate the decision-making problem with a finite set of alternatives,in which the decision information takes the form of a fuzzy preference relation. We develop asimple and practical approach to obtaining the priority vector of a fuzzy preference relation. Theprominent characteristic of the developed approach is that the priority vector can generally beobtained by a simple formula, which is derived from a quadratic programming model. We utilize theconsistency ratio to check the consistency of fuzzy preference relation. If the fuzzy preferencerelation is of unacceptable consistency, then we can return it to the decision maker to reconsiderstructuring a new fuzzy preference relation until the fuzzy preference relation with acceptableconsistency is obtained. We finally illustrate the priority approach by two numerical examples. Thenumerical results show that the developed approach is straightforward, effective, and can easily beperformed on a computer.

A POLYNOMIAL PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one corrector step after each predictor step, where Step 2 is a predictor step and Step 4 is a corrector step in the algorithm. In the algorithm, the predictor step decreases the dual gap as much as possible in a wider neighborhood of the central path and the corrector step draws iteration points back to a narrower neighborhood and make a reduction for the dual gap. It is shown that the algorithm has O（√nL） iteration complexity which is the best result for convex quadratic programming so far.

A COMBINED PARAMETRIC QUADRATIC PROGRAMMING AND PRECISE INTEGRATION METHOD BASED DYNAMIC ANALYSIS OF ELASTIC-PLASTIC HARDENING/SOFTENING PROBLEMS

The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcome the result mesh-sensitivity problem in the dynamic strain softening or strain localization analysis. The equations for the dynamic elastic-plastic problems are derived in terms of the parametric variational principle, which is valid for associated, non-associated and strain softening plastic constitutive models in the finite element analysis. The precise integration method, which has been widely used for discretization in time domain of the linear problems, is introduced for the solution of dynamic nonlinear equations. The new algorithm proposed is based on the combination of the parametric quadratic programming method and the precise integration method and has all the advantages in both of the algorithms. Results of numerical examples demonstrate not only the validity, but also the advantages of the algorithm proposed for the numerical solution of nonlinear dynamic problems.

Orthogonal genetic algorithm for solving quadratic bilevel programming problems

A quadratic bilevel programming problem is transformed into a single level complementarity slackness problem by applying Karush-Kuhn-Tucker（KKT） conditions.To cope with the complementarity constraints,a binary encoding scheme is adopted for KKT multipliers,and then the complementarity slackness problem is simplified to successive quadratic programming problems,which can be solved by many algorithms available.Based on 0-1 binary encoding,an orthogonal genetic algorithm,in which the orthogonal experimental design with both two-level orthogonal array and factor analysis is used as crossover operator,is proposed.Numerical experiments on 10 benchmark examples show that the orthogonal genetic algorithm can find global optimal solutions of quadratic bilevel programming problems with high accuracy in a small number of iterations.

Sequential quadratic programming-based non-cooperative target distributed hybrid processing optimization method

The distributed hybrid processing optimization problem of non-cooperative targets is an important research direction for future networked air-defense and anti-missile firepower systems. In this paper, the air-defense anti-missile targets defense problem is abstracted as a nonconvex constrained combinatorial optimization problem with the optimization objective of maximizing the degree of contribution of the processing scheme to non-cooperative targets, and the constraints mainly consider geographical conditions and anti-missile equipment resources. The grid discretization concept is used to partition the defense area into network nodes, and the overall defense strategy scheme is described as a nonlinear programming problem to solve the minimum defense cost within the maximum defense capability of the defense system network. In the solution of the minimum defense cost problem, the processing scheme, equipment coverage capability, constraints and node cost requirements are characterized, then a nonlinear mathematical model of the non-cooperative target distributed hybrid processing optimization problem is established, and a local optimal solution based on the sequential quadratic programming algorithm is constructed, and the optimal firepower processing scheme is given by using the sequential quadratic programming method containing non-convex quadratic equations and inequality constraints. Finally, the effectiveness of the proposed method is verified by simulation examples.

An Interval Maximum Entropy Method for Quadratic Programming Problem

With the idea of maximum entropy function and penalty function methods, we transform the quadratic programming problem into an unconstrained differentiable optimization problem, discuss the interval extension of the maximum entropy function, provide the region deletion test rules and design an interval maximum entropy algorithm for quadratic programming problem. The convergence of the method is proved and numerical results are presented. Both theoretical and numerical results show that the method is reliable and efficient.

A Quadratic Programming Model for Blast Scheduling

A quadratic programming model is established to choose the blocks to be blasted in a given period. The length of this period depends on the production planning requirements. During the given period, the blocks' parameters are available from the geological database of the mine. The objective is to minimize the deviation of the average ore grade of blasted blocks from the standard ore grade required by the mill. Transportation ability constraint. production quantity demand constraint. minimum safety bench constraint. block size constraint and block, bench precedence constraints are considered in forming the programming model. This model has more practical objective function and reasonable constraints compared with the existing model for this kind of problems.

MIXED ENERGY METHOD FOR SOLUTION OF QUADRATIC PROGRAMMING PROBLEMS AND ELASTIC-PLASTIC ANALYSIS OF TRUSS STRUCTURES

A new algorithm for the solution of quadratic programming problemsis put forward in terms of the mixed energy theory and is furtherused for the incremental solution of elastic-plastic trussstructures. The method proposed is different from the traditionalone, for which the unknown variables are selected just in one classsuch as displacements or stresses. The present method selects thevariables in the mixed form with both displacement and stress. As themethod is established in the hybrid space, the information found inthe previous incremental step can be used for the solution of thepresent step, making the algorithm highly effi- cient in thenumerical solution process of quadratic programming problems. Theresults obtained in the exm- ples of the elastic-plastic solution ofthe truss structures verify what has been predicted in thetheoretical anal- ysis.

Genetic Algorithm for Solving Quadratic Bilevel Programming Problem

By applying Kuhn-Tucker condition the quadratic bilevel programming, a class of bilevel programming, is transformed into a single level programming problem, which can be simplified by some rule. So we can search the optimal solution in the feasible region, hence reduce greatly the searching space. Numerical experiments on several literature problems show that the new algorithm is both feasible and effective in practice.

Global optimality conditions for quadratic 0-1 programming with inequality constraints

Quadratic 0-1 problems with linear inequality constraints are briefly considered in this paper.Global optimality conditions for these problems,including a necessary condition and some sufficient conditions,are presented.The necessary condition is expressed without dual variables.The relations between the global optimal solutions of nonconvex quadratic 0-1 problems and the associated relaxed convex problems are also studied.

A quadratic programming method for optimal degree reduction of Bézier curves with G^1-continuity

This paper presents a quadratic programming method for optimal multi-degree reduction of B6zier curves with G^1-continuity. The L2 and I2 measures of distances between the two curves are used as the objective functions. The two additional parameters, available from the coincidence of the oriented tangents, are constrained to be positive so as to satisfy the solvability condition. Finally, degree reduction is changed to solve a quadratic problem of two parameters with linear constraints. Applications of degree reduction of Bezier curves with their parameterizations close to arc-length parameterizations are also discussed.

Quadratic Programming-based Approach for Autonomous Vehicle Path Planning in Space

Path planning for space vehicles is still a challenging problem although considerable progress has been made over the past decades.The major difficulties are that most of existing methods only adapt to static environment instead of dynamic one,and also can not solve the inherent constraints arising from the robot body and the exterior environment.To address these difficulties,this research aims to provide a feasible trajectory based on quadratic programming（QP） for path planning in three-dimensional space where an autonomous vehicle is requested to pursue a target while avoiding static or dynamic obstacles.First,the objective function is derived from the pursuit task which is defined in terms of the relative distance to the target,as well as the angle between the velocity and the position in the relative velocity coordinates（RVCs）.The optimization is in quadratic polynomial form according to QP formulation.Then,the avoidance task is modeled with linear constraints in RVCs.Some other constraints,such as kinematics,dynamics,and sensor range,are included.Last,simulations with typical multiple obstacles are carried out,including in static and dynamic environments and one of human-in-the-loop.The results indicate that the optimal trajectories of the autonomous robot in three-dimensional space satisfy the required performances.Therefore,the QP model proposed in this paper not only adapts to dynamic environment with uncertainty,but also can satisfy all kinds of constraints,and it provides an efficient approach to solve the problems of path planning in three-dimensional space.

Continuity of the optimal value function and optimal solutions of parametric mixed-integer quadratic programs

To properly describe and solve complex decision problems, research on theoretical properties and solution of mixed-integer quadratic programs is becoming very important. We establish in this paper different Lipschitz-type continuity results about the optimal value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of the linear constraints. The obtained results extend some existing results for continuous quadratic programs, and, more importantly, lay the foundation for further theoretical study and corresponding algorithm analysis on mixed-integer quadratic programs.

Lipschitz continuity of the optimal value function and KKT solution set in indefinite quadratic programs

When all the involved data in indefinite quadratic programs change simultaneously, we show the locally Lipschtiz continuity of the KKT set of the quadratic programming problem firstly, then we establish the locally Lipschtiz continuity of the KKT solution set. Finally, the similar conclusion for the corresponding optimal value function is obtained.

Successive quadratic programming multiuser detector

Based on the semidefinite programming relaxation of the CDMA maximum likelihood multiuser detection problem, a detection strategy by the successive quadratic programming algorithm is presented. Coupled with the randomized cut generation scheme, the suboptimal solution of the multiuser detection problem in obtained. Compared to the interior point methods previously reported based on semidefmite programming, simulations demonstrate that the successive quadratic programming algorithm often yields the similar BER performances of the multiuser detection problem. But the average CPU time of this approach is significantly reduced.

Automatic differentiation for reduced sequential quadratic programming

In order to slove the large-scale nonlinear programming （NLP） problems efficiently, an efficient optimization algorithm based on reduced sequential quadratic programming （rSQP） and automatic differentiation （AD） is presented in this paper. With the characteristics of sparseness, relatively low degrees of freedom and equality constraints utilized, the nonlinear programming problem is solved by improved rSQP solver. In the solving process, AD technology is used to obtain accurate gradient information. The numerical results show that the combined algorithm, which is suitable for large-scale process optimization problems, can calculate more efficiently than rSQP itself.

Checking weak and strong optimality of the solution to interval convex quadratic program

In this paper,we investigate three canonical forms of interval convex quadratic pro-gramming problems.Necessary and suficient conditions for checking weak and strong optimality of given vector corresponding to various forms of feasible region,are established respectively.By using the concept of feasible direction,these conditions are formulated in the form of linear systems with both equations and inequalities.In addition,we provide two specific examples to illustrate the efficiency of the conditions.

AN INEXACT LAGRANGE-NEWTON METHOD FOR STOCHASTIC QUADRATIC PROGRAMS WITH RECOURSE

In this paper,two-stage stochastic quadratic programming problems with equality constraints are considered.By Monte Carlo simulation-based approximations of the objective function and its first(second)derivative,an inexact Lagrange-Newton type method is proposed.It is showed that this method is globally convergent with probability one.In particular,the convergence is local superlinear under an integral approximation error bound condition.Moreover,this method can be easily extended to solve stochastic quadratic programming problems with inequality constraints.

SEQUENTIAL QUADRATIC PROGRAMMING METHODS FOR OPTIMAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

A kind of direct methods is presented for the solution of optimal control problems with state constraints. These methods are sequential quadratic programming methods. At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and linear approximations to constraints is solved to get a search direction for a merit function. The merit function is formulated by augmenting the Lagrangian function with a penalty term. A line search is carried out along the search direction to determine a step length such that the merit function is decreased. The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadratic programming methods.