Variational Problems Involving a Caputo-Type Fractional Derivative

The aim of this paper is to study certain problems of calculus of variations that are dependent upon a Lagrange function on a Caputo-type fractional derivative. This type of fractional operator is a generalization of the Caputo and the Caputo–Hadamard fractional derivatives that are dependent on a real parameter \(\rho \). Sufficient and necessary conditions of the first and second order are presented. The cases of integral and holonomic constraints are also considered.     

Fractional variational integrators for fractional Euler–Lagrange equations with holonomic constraints

In this paper, the fractional variational integrators developed by Wang and Xiao (2012) [28] are extended to the fractional Euler–Lagrange (E–L) equations with holonomic constraints. The corresponding fractional discrete E–L equations are derived, and their local convergence is discussed. Some fractional variational integrators are presented. The suggested methods are shown to be efficient by some numerical examples.     

Lagrange Multiplier Conditions Characterizing the Optimal Solution Sets of Cone-Constrained Convex Programs

Various characterizations of optimal solution sets of cone-constrained convex optimization problems are given. The results are expressed in terms of subgradients and Lagrange multipliers. We establish first that the Lagrangian function of a convex program is constant on the optimal solution set. This elementary property is then used to derive various simple Lagrange multiplier-based characterizations of the solution set. For a finite-dimensional convex program with inequality constraints, the characterizations illustrate that the active constraints with positive Lagrange multipliers at an optimal solution remain active at all optimal solutions of the program. The results are applied to derive corresponding Lagrange multiplier characterizations of the solution sets of semidefinite programs and fractional programs. Specific examples are given to illustrate the nature of the results.     

Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.     

Fractional variational problems depending on indefinite integrals

We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and natural boundary conditions, which provide a generalization of the previous results found in the literature. Isoperimetric problems, problems with holonomic constraints and depending on higher-order Caputo derivatives, as well as fractional Lagrange problems, are considered.     

Optimal control problem with an integral equation as the control object

We consider a nonlinear optimal control problem with an integral equation as the control object, subject to control constraints. This integral equation corresponds to the fractional moment of a stochastic process involving short-range and long-range dependences. For both cases, we derive the first-order necessary optimality conditions in the form of the Euler–Lagrange equation, and then apply them to obtain a numerical solution of the problem of optimal portfolio selection.     

对合变换和薄板弯曲问题的多变量变分原理

本文利用拉氏乘子法把薄板弯曲问题的最小位能原理和最小余能原理的变分约束条件解除.从而导出了常见的广义变分原理.为了降低泛函中变量导数的阶次.我们用对合变换引进新的正则变量.于是,我们可以进一步利用拉氏乘子法,把这些对合变换当作变分约束而予以消除,从而导出了各种多变量的薄板弯曲广义变分原理.事实证明,使用上述拉氏乘子法,并不能消除一切变分约束;为此,我们进一步引用高阶拉氏乘子法消除这些剩下来的约束条件,从而导得了薄板弯曲问题的更一般的广义变分原理.     

About fractional quantization and fractional variational principles

In this paper, a new method of finding the fractional Euler–Lagrange equations within Caputo derivative is proposed by making use of the fractional generalization of the classical Faá di Bruno formula. The fractional Euler–Lagrange and the fractional Hamilton equations are obtained within the 1 + 1 field formalism. One illustrative example is analyzed.     

A Pseudospectral Method for Fractional Optimal Control Problems

In this article, a direct pseudospectral method based on Lagrange interpolating functions with fractional power terms is used to solve the fractional optimal control problem. As most applied fractional problems have solutions in terms of the fractional power, using appropriate characteristic nodal-based functions with suitable power leads to a more accurate pseudospectral approximation of the solution. The Lagrange interpolating functions and their fractional derivatives belong to the Müntz space; such functions are employed to show that a relationship exists between the Karush–Kukn–Tucker conditions associated with nonlinear programming and the first optimal necessary conditions. Furthermore, the convergence of the method is investigated. The obtained numerical results are an indication of the behavior of the algorithm.     

Euler–Lagrange Equations for Lagrangians Containing Complex-order Fractional Derivatives

Two variational problems of finding the Euler–Lagrange equations corresponding to Lagrangians containing fractional derivatives of real- and complex-order are considered. The first one is the unconstrained variational problem, while the second one is the fractional optimal control problem. The expansion formula for fractional derivatives of complex-order is derived in order to approximate the fractional derivative appearing in the Lagrangian. As a consequence, a sequence of approximated Euler–Lagrange equations is obtained. It is shown that the sequence of approximated Euler–Lagrange equations converges to the original one in the weak sense as well as that the sequence of the minimal values of approximated action integrals tends to the minimal value of the original one.     

Variational iteration method for the Burgers’ flow with fractional derivatives—New Lagrange multipliers

The flow through porous media can be better described by fractional models than the classical ones since they include inherently memory effects caused by obstacles in the structures. The variational iteration method was extended to find approximate solutions of fractional differential equations with the Caputo derivatives, but the Lagrange multipliers of the method were not identified explicitly. In this paper, the Lagrange multiplier is determined in a more accurate way and some new variational iteration formulae are presented.     

一类带反凸约束的非线性比式和问题的全局优化算法

本文针对一类带有反凸约束的非线性比式和分式规划问题,提出一种求其全局最优解的单纯形分支和对偶定界算法.该算法利用Lagrange对偶理论将其中关键的定界问题转化为一系列易于求解的线性规划问题.收敛性分析和数值算例均表明提出的算法是可行的.     

The DuBois–Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler–Lagrange Equation Involving Only Derivatives of Caputo

Derivatives and integrals of noninteger order were introduced more than three centuries ago but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus fail to give an Euler–Lagrange equation with only Caputo derivatives. In this work, we propose a new approach to the fractional calculus of variations by generalizing the DuBois–Reymond lemma and showing how Euler–Lagrange equations involving only Caputo derivatives can be obtained.     

Stable iterative Lagrange principle in convex programming as a tool for solving unstable problems

A convex programming problem in a Hilbert space with an operator equality constraint and a finite number of functional inequality constraints is considered. All constraints involve parameters. The close relation of the instability of this problem and, hence, the instability of the classical Lagrange principle for it to its regularity properties and the subdifferentiability of the value function in the problem is discussed. An iterative nondifferential Lagrange principle with a stopping rule is proved for the indicated problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimization problems is discussed. The capabilities of this principle are illustrated by numerically solving the classical ill-posed problem of finding the normal solution of a Fredholm integral equation of the first kind.     

A modified objective function method for solving nonlinear multiobjective fractional programming problems

A new method is used for solving nonlinear multiobjective fractional programming problems having V-invex objective and constraint functions with respect to the same function η. In this approach, an equivalent vector programming problem is constructed by a modification of the objective fractional function in the original nonlinear multiobjective fractional problem. Furthermore, a modified Lagrange function is introduced for a constructed vector optimization problem. By the help of the modified Lagrange function, saddle point results are presented for the original nonlinear fractional programming problem with several ratios. Finally, a Mond-Weir type dual is associated, and weak, strong and converse duality results are established by using the introduced method with a modified function. To obtain these duality results between the original multiobjective fractional programming problem and its original Mond-Weir duals, a modified Mond-Weir vector dual problem with a modified objective function is constructed.     

Formulation of Euler-Lagrange equations for fractional variational problems

This paper presents extensions to traditional calculus of variations for systems containing fractional derivatives. The fractional derivative is described in the Riemann-Liouville sense. Specifically, we consider two problems, the simplest fractional variational problem and the fractional variational problem of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives and unknown functions. For the second problem, we also present a Lagrange type multiplier rule. For both problems, we develop the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Two problems are considered to demonstrate the application of the formulation. The formulation presented and the resulting equations are very similar to those that appear in the field of classical calculus of variations.     

两个泛函优化定理的证明

在泛函优化理论中,Lagrange乘子定理、对偶定理占有重要地位.建立了带有等式和不等式约束的泛函优化问题,并给出了广义Lagrange乘子定理、广义Lagrange对偶定理的证明.     

An approach to find redundant objective function(s) and redundant constraint(s) in multi-objective nonlinear stochastic fractional programming problems

Structural redundancies in mathematical programming models are nothing uncommon and nonlinear programming problems are no exception. Over the past few decades numerous papers have been written on redundancy. Redundancy in constraints and variables are usually studied in a class of mathematical programming problems. However, main emphasis has so far been given only to linear programming problems. In this paper, an algorithm that identifies redundant objective function(s) and redundant constraint(s) simultaneously in multi-objective nonlinear stochastic fractional programming problems is provided. A solution procedure is also illustrated with numerical examples. The proposed algorithm reduces the number of nonlinear fractional objective functions and constraints in cases where redundancy exists.     

高阶拉氏乘子法和弹性理论中更一般的广义变分原理

作者曾指出[1],弹性理论的最小位能原理和最小余能原理都是有约束条件限制下的变分原理采用拉格朗日乘子法,我们可以把这些约束条件乘上待定的拉氏乘子,计入有关变分原理的泛函内,从而将这些有约束条件的极值变分原理,化为无条件的驻值变分原理.如果把这些待定拉氏乘子和原来的变量都看作是独立变量而进行变分,则从有关泛函的驻值条件就可以求得这些拉氏乘子用原有物理变量表示的表达式.把这些表达式代入待定的拉氏乘子中,即可求所谓广义变分原理的驻值变分泛函.但是某些情况下,待定的拉氏乘子在变分中证明恒等于零.这是一种临界的变分状态.在这种临界状态中,我们无法用待定拉氏乘子法把变分约束条件吸收入泛函,从而解除这个约束条件.从最小余能原理出发,利用待定拉氏乘子法,企图把应力应变关系这个约束条件吸收入有关泛函时,就发生这种临界状态,用拉氏乘子法,从余能原理只能导出Hellinger-Reissner变分原理[2],[3],这个原理中只有应力和位移两类独立变量,而应力应变关系则仍是变分约束条件,人们利用这个条件,从变分求得的应力中求应变.所以Hellinger-Reissner变分原理仍是一种有条件的变分原理.     

Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions

In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic optimization problems. The work of Z. Y. Wu was carried out while the author was at the Department of Applied Mathematics, University of New South Wales, Sydney, Australia.