The Fritz John and Kuhn-Tucker optimality conditions in continuous-time nonlinear programming

First-order stationary-point necessary optimality criteria of both the Fritz John and Kuhn-Tucker type are established for continuous-time nonlinear programming problems. Furthermore, the relationship between these criteria and saddlepoint optimality conditions is also discussed. The main auxiliary result employed in the derivation of the principal optimality criteria is a continuous-time version of Gordan's transposition theorem.     

Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems

A Dinkelbach-type algorithm is proposed in this paper to solve a class of continuous-time linear fractional programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.     

Using the parametric approach to solve the continuous-time linear fractional max–min problems

A numerical algorithm based on parametric approach is proposed in this paper to solve a class of continuous-time linear fractional max-min programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as a parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this algorithm.     

Application of continuous-time programming problems to a class of variational-type inequalities

In this paper, we establish sufficiency criteria under generalized ρ−(η,θ)-invexity conditions for general continuous-time programming problems with nonlinear equality/inequality constraints. Using this we establish some existence criteria for solutions of a class of variational-type inequalities.     

Optimality conditions and Lagrangian duality in continuous-time nonlinear programming

Using a perturbation approach, the Kuhn-Tucker saddlepoint and stationary-point optimality conditions and a Lagrangian duality theory are established for a general class of continuous-time nonlinear programming problems. It is shown that most of the duality formulations in the existing literature of continuous programming are special cases of this Lagrangian formulation.     

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR A CLASS OF NONSMOOTH CONTINUOUS-TIME GENERALIZED FRACTIONAL PROGRAMMING PROBLEMS

Abstract

Both parametric and parameter-free stationary-point-type and saddle-point-type necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time generalized fractional programming problems with Volterra-type integral inequality and nonnegativity constraints. These optimality criteria are then utilized for constructing ten parametric and parameter-free Wolfe-type and Lagrangian-type dual problems and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be obtained for two important special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. All the results developed here are also applicable to continuous-time programming problems with fractional, discrete max, and conventional objective functions, which are special cases of the main problem studied in this paper.     

Continuous-Time Generalized Fractional Programming Problems. Part I: Basic Theory

This study, that will be presented as two parts, develops a computational approach to a class of continuous-time generalized fractional programming problems. The parametric method for finite-dimensional generalized fractional programming is extended to problems posed in function spaces. The developed method is a hybrid of the parametric method and discretization approach. In this paper (Part I), some properties of continuous-time optimization problems in parametric form pertaining to continuous-time generalized fractional programming problems are derived. These properties make it possible to develop a computational procedure for continuous-time generalized fractional programming problems. However, it is notoriously difficult to find the exact solutions of continuous-time optimization problems. In the accompanying paper (Part II), a further computational procedure with approximation will be proposed. This procedure will yield bounds on errors introduced by the numerical approximation. In addition, both the size of discretization and the precision of an approximation approach depend on predefined parameters.     

Continuous-time optimization problems involving invex functions

In [D.H. Martin, The essence of invexity, J. Optim. Theory Appl. 47 (1985) 65-76] Martin introduced the notions of KKT-invexity and WD-invexity for mathematical programming problems. These notions are relaxations of invexity. In this work we generalize these concepts for continuous-time nonlinear optimization problems. We prove that the notion of KKT-invexity is a necessary and sufficient condition for global optimality of a Karush-Kuhn-Tucker point and that the notion of WD-invexity is a necessary and sufficient condition for weak duality.     

Optimality criteria in E-convex programming

Necessary and sufficient optimality criteria in nonlinear programming are discussed for a class of E-convex programming problems which is considered more general than a class of convex programming problems. We also modify the Fritz John and the Kuhn–Tucker problems to E-Fritz John and E-Kuhn–Tucker problems.     

On Applied Nonlinear and Bilevel Programming or Pursuit-Evasion Games

Motivated by the benefits of discretization in optimal control problems, we consider the possibility of discretizing pursuit-evasion games. Two approaches are introduced. In the first approach, the solution of the necessary conditions of the continuous-time game is decomposed into ordinary optimal control problems that can be solved using discretization and nonlinear programming techniques. In the second approach, the game is discretized and transformed into a bilevel programming problem, which is solved using a first-order feasible direction method. Although the starting points of the approaches are different, they lead in practice to the same solution algorithm. We demonstrate the usability of the discretization by solving some open-loop representations of feedback solutions for a complex pursuit-evasion game between a realistically modeled aircraft and a missile, with terminal time as the payoff. The solutions are compared with those obtained via an indirect method.     

Continuous-time linear programming problems revisited: A perturbation approach

The class of continuous-time linear programming problems under the assumption that the constraints are satisfied almost everywhere in the time interval [0,?T]?is taken into account in this article. Under this assumption, its corresponding discretized problems cannot be formulated by equally dividing the time interval [0,?T]?as subintervals of [0,?T]?. In this article, we also introduce the perturbed continuous-time linear programming problems to prove the strong duality theorem when the constraints are assumed to be satisfied a.e. in [0,?T]?.     

Approximate Solutions and Duality Theorems for Continuous-Time Linear Fractional Programming Problems

This article proposes a practical computational procedure to solve a class of continuous-time linear fractional programming problems by designing a discretized problem. Using the optimal solutions of proposed discretized problems, we construct a sequence of feasible solutions of continuous-time linear fractional programming problem and show that there exists a subsequence that converges weakly to a desired optimal solution. We also establish an estimate of the error bound. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.     

Optimality criteria in nonlinear programming involving nonconvex functions

In this paper a new class of functions, called η-convex, η-quasiconvex and η-pseudoconvex, are defined and a number of sufficient optimality criteria are obtained for nonlinear programming problems involving these functions.     

A NEW INEXACT SEQUENTIAL QUADRATIC PROGRAMMING ALGORITHM

This paper represents an inexact sequential quadratic programming (SQP) algorithm which can solve nonlinear programming (NLP) problems. An inexact solution of the quadratic programming subproblem is determined by a projection and contraction method such that only matrix-vector product is required. Some truncated criteria are chosen such that the algorithm is suitable to large scale NLP problem. The global convergence of the algorithm is proved.     

Efficient dynamic programming implementations of Newton's method for unconstrained optimal control problems

Naive implementations of Newton's method for unconstrainedN-stage discrete-time optimal control problems with Bolza objective functions tend to increase in cost likeN ³ asN increases. However, if the inherent recursive structure of the Bolza problem is properly exploited, the cost of computing a Newton step will increase only linearly withN. The efficient Newton implementation scheme proposed here is similar to Mayne's DDP (differential dynamic programming) method but produces the Newton step exactly, even when the dynamical equations are nonlinear. The proposed scheme is also related to a Riccati treatment of the linear, two-point boundary-value problems that characterize optimal solutions. For discrete-time problems, the dynamic programming approach and the Riccati substitution differ in an interesting way; however, these differences essentially vanish in the continuous-time limit.This work was supported by the National Science Foundation, Grant No. DMS-85-03746.     

用非线性方程组求解等式约束非线性规划问题的降维算法

本文研究线性和非线性等式约束非线性规划问题的降维算法.首先,利用一般等式约束问题的降维方法,将线性等式约束非线性规划问题转换成一个非线性方程组,解非线性方程组即得其解;然后,对线性和非线性等式约束非线性规划问题用Lagrange乘子法,将非线性约束部分和目标函数构成增广的Lagrange函数,并保留线性等式约束,这样便得到一个线性等式约束非线性规划序列,从而,又将问题转化为求解只含线性等式约束的非线性规划问题.     

Note: continuous-time linear programming problems revisited

This note is aimed to correct the strong duality theorem of previous paper regarding the continuous-time linear programming problems. The argument presented in the previous paper can only be used to prove the case of piecewise continuous functions in which the discontinuities are the left-continuities.     

不完全拉格朗日函数和具有(F α,ρ,d)凸性的数学规划的鞍点判别准则

用一个不完全拉格朗日函数研究一类具有(F,α,ρ,d)凸性假设的非线性规划问题的鞍点最优性判别准则,为了得到最优解与鞍点之间的关系,给出了(F,α,ρ,d)凸性中参数所要满足的条件．     

A sparse nonlinear optimization algorithm

One of the most effective numerical techniques for solving nonlinear programming problems is the sequential quadratic programming approach. Many large nonlinear programming problems arise naturally in data fitting and when discretization techniques are applied to systems described by ordinary or partial differential equations. Problems of this type are characterized by matrices which are large and sparse. This paper describes a nonlinear programming algorithm which exploits the matrix sparsity produced by these applications. Numerical experience is reported for a collection of trajectory optimization problems with nonlinear equality and inequality constraints.The authors wish to acknowledge the insightful contributions of Dr. William Huffman.     

不完全拉格朗日函数和具有（F，α,ρ，d）凸性的数学规划的鞍点判别准则

用一个不完全拉格朗日函数研究一类具有（F,α,ρ，d）凸性假设的非线性规划问题的鞍点最优性判别准则，为了得到最优解与鞍点之间的关系，给出了（F,α,ρ，d）凸性中参数所要满足的条件．