Solving Fractional Multicriteria Optimization Problems with Sum of Squares Convex Polynomial Data

This paper focuses on the study of finding efficient solutions in fractional multicriteria optimization problems with sum of squares convex polynomial data. We first relax the fractional multicriteria optimization problems to fractional scalar ones. Then, using the parametric approach, we transform the fractional scalar problems into non-fractional problems. Consequently, we prove that, under a suitable regularity condition, the optimal solution of each non-fractional scalar problem can be found by solving its associated single semidefinite programming problem. Finally, we show that finding efficient solutions in the fractional multicriteria optimization problems is tractable by employing the epsilon constraint method. In particular, if the denominators of each component of the objective functions are same, then we observe that efficient solutions in such a problem can be effectively found by using the hybrid method. Some numerical examples are given to illustrate our results.     

A Combination of Variational and Penalty Methods for Solving a Class of Fractional Optimal Control Problems

This paper develops an approximate method, based on the combination of epsilon penalty and variational methods, for solving a class of multidimensional fractional optimal control problems. The fractional derivative is in the Caputo sense. In the presented method, utilizing the epsilon method, the given optimal control problem transforms into an unconstrained optimization problem; then, the equivalent variational equality is derived for the given unconstrained problem. The variational equality is approximately solved by applying a spectral method.     

Epsilon-Ritz Method for Solving a Class of Fractional Constrained Optimization Problems

In this paper, epsilon and Ritz methods are applied for solving a general class of fractional constrained optimization problems. The goal is to minimize a functional subject to a number of constraints. The functional and constraints can have multiple dependent variables, multiorder fractional derivatives, and a group of initial and boundary conditions. The fractional derivative in the problem is in the Caputo sense. The constrained optimization problems include isoperimetric fractional variational problems (IFVPs) and fractional optimal control problems (FOCPs). In the presented approach, first by implementing epsilon method, we transform the given constrained optimization problem into an unconstrained problem, then by applying Ritz method and polynomial basis functions, we reduce the optimization problem to the problem of optimizing a real value function. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. The convergence of the method is analytically studied and some illustrative examples including IFVPs and FOCPs are presented to demonstrate validity and applicability of the new technique.     

A numerical technique for solving fractional variational problems

This paper presents an accurate numerical method for solving a class of fractional variational problems (FVPs). The fractional derivative in these problems is in the Caputo sense. The proposed method is called fractional Chebyshev finite difference method. In this technique, we approximate FVPs and end up with a finite‐dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. The fractional derivative approximation using Clenshaw and Curtis formula introduced here, along with Clenshaw and Curtis procedure for the numerical integration of a non‐singular functions and the Rayleigh–Ritz method for the constrained extremum, is considered. By this method, the given problem is reduced to the problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of FVPs. Special attention is given to study the convergence analysis and evaluate an error upper bound of the obtained approximate formula. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique. A comparison with another method is given. Copyright © 2012 John Wiley & Sons, Ltd.     

Global optimization of polynomial-expressed nonlinear optimal control problems with semidefinite programming relaxation

In this paper, we propose a new deterministic global optimization method for solving nonlinear optimal control problems in which the constraint conditions of differential equations and the performance index are expressed as polynomials of the state and control functions. The nonlinear optimal control problem is transformed into a relaxed optimal control problem with linear constraint conditions of differential equations, a linear performance index, and a matrix inequality condition with semidefinite programming relaxation. In the process of introducing the relaxed optimal control problem, we discuss the duality theory of optimal control problems, polynomial expression of the approximated value function, and sum-of-squares representation of a non-negative polynomial. By solving the relaxed optimal control problem, we can obtain the approximated global optimal solutions of the control and state functions based on the degree of relaxation. Finally, the proposed global optimization method is explained, and its efficacy is proved using an example of its application.     

A numerical technique for solving a class of fractional variational problems

This paper presents a numerical method for solving a class of fractional variational problems (FVPs) with multiple dependent variables, multi order fractional derivatives and a group of boundary conditions. The fractional derivative in the problem is in the Caputo sense. In the presented method, the given optimization problem reduces to a system of algebraic equations using polynomial basis functions. An approximate solution for the FVP is achieved by solving the system. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique.     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.     

A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems

We develop a simple and accurate method to solve fractional variational and fractional optimal control problems with dependence on Caputo and Riemann–Liouville operators. Using known formulas for computing fractional derivatives of polynomials, we rewrite the fractional functional dynamical optimization problem as a classical static optimization problem. The method for classical optimal control problems is called Ritz’s method. Examples show that the proposed approach is more accurate than recent methods available in the literature.     

Identification of constitutive parameters for fractional viscoelasticity

This paper develops a numerical model to identify constitutive parameters in the fractional viscoelastic field. An explicit semi-analytical numerical model and a finite difference (FD) method based numerical model are derived for solving the direct homogenous and regionally inhomogeneous fractional viscoelastic problems, respectively. A continuous ant colony optimization (ACO) algorithm is employed to solve the inverse problem of identification. The feasibility of the proposed approach is illustrated via the numerical verification of a two-dimensional identification problem formulated by the fractional Kelvin–Voigt model, and the noisy data and regional inhomogeneity etc. are taken into account.     

New analysis and application of fractional order Schrödinger equation using with Atangana–Batogna numerical scheme

In this work, an analytical approximation to the solution of Schrodinger equation has been provided. The fractional derivative used in this equation is the Caputo derivative. The existence and uniqueness conditions of solutions for the proposed model are derived based on the power law. While solving the fractional order Schrodinger equation, Atangana–Batogna numerical method is presented for fractional order equation. We obtain an efficient recurrence relation for solving these kinds of equations. To illustrate the usefulness of the numerical scheme, the numerical simulations are presented. The results show that the numerical scheme is very effective and simple.     

Robust truss topology optimization via semidefinite programming with complementarity constraints: a difference-of-convex programming approach

The robust truss topology optimization against the uncertain static external load can be formulated as mixed-integer semidefinite programming. Although a global optimal solution can be computed with a branch-and-bound method, it is very time-consuming. This paper presents an alternative formulation, semidefinite programming with complementarity constraints, and proposes an efficient heuristic. The proposed method is based upon the concave–convex procedure for difference-of-convex programming. It is shown that the method can often find a practically reasonable truss design within the computational cost of solving some dozen of convex optimization subproblems.     

稠密k-子图问题的双非负松弛

稠密k-子图问题是组合优化里面一类经典的优化问题,其在通常情况下是非凸且NP-难的。本文给出了求解该问题的一个新凸松弛方法-双非负松弛方法,并建立了问题的相应双非负松弛模型,而且证明了其在一定的条件下等价于一个新的半定松弛模型。最后,我们使用一些随机例子对这些模型进行了数值测试,测试的结果表明双非负松弛的计算效果要优于等价的半定松弛。     

Numerical solution of optimization problems in vibrational mechanics using the characteristics method

Optimal control problems with a terminal pay-off functional are considered. The dynamics of the control system consists of rapid oscillatory and slow non-linear motions. A numerical method for solving these problems using the characteristics of the Hamilton–Jacobi–Bellman equation is presented. Estimates of the accuracy of the method are obtained. A theorem is proved which enables one to determine the class of functions containing the optimal preset control to be obtained. The results of the numerical solution of a terminal optimization problem for a fast non-linear pendulum are presented.     

Finding Robust Global Optimal Values of Bilevel Polynomial Programs with Uncertain Linear Constraints

This paper studies a bilevel polynomial program involving box data uncertainties in both its linear constraint set and its lower-level optimization problem. We show that the robust global optimal value of the uncertain bilevel polynomial program is the limit of a sequence of values of Lasserre-type hierarchy of semidefinite linear programming relaxations. This is done by first transforming the uncertain bilevel polynomial program into a single-level non-convex polynomial program using a dual characterization of the solution of the lower-level program and then employing the powerful Putinar’s Positivstellensatz of semi-algebraic geometry. We provide a numerical example to show how the robust global optimal value of the uncertain bilevel polynomial program can be calculated by solving a semidefinite programming problem using the MATLAB toolbox YALMIP.     

Finite difference/$H^1$-Galerkin MFE procedure for a fractional water wave model

In this article, an $H^1$-Galerkin mixed finite element (MFE) method for solving the time fractional water wave model is presented. First-order backward Euler difference method and $L1$ formula are applied to approximate integer derivative and Caputo fractional derivative with order $1/2$, respectively, and $H^1$-Galerkin mixed finite element method is used to approximate the spatial direction. The analysis of stability for fully discrete mixed finite element scheme is made and the optimal space-time orders of convergence for two unknown variables in both $H^1$-norm and $L^2$-norm are derived. Further, some computing results for a priori analysis and numerical figures based on four changed parameters in the studied problem are given to illustrate the effectiveness of the current method     

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.     

Fractional-Order Legendre Functions and Their Application to Solve Fractional Optimal Control of Systems Described by Integro-differential Equations

In this paper, we introduce a set of functions called fractional-order Legendre functions (FLFs) to obtain the numerical solution of optimal control problems subject to the linear and nonlinear fractional integro-differential equations. We consider the properties of these functions to construct the operational matrix of the fractional integration. Also, we achieved a general formulation for operational matrix of multiplication of these functions to solve the nonlinear problems for the first time. Then by using these matrices the mentioned fractional optimal control problem is reduced to a system of algebraic equations. In fact the functions of the problem are approximated by fractional-order Legendre functions with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem converts to an optimization problem, which can then be solved numerically. The convergence of the method is discussed and finally, some numerical examples are presented to show the efficiency and accuracy of the method.     

Eigenvalue Multiplicity Estimate in Semidefinite Programming

A semidefinite programming problem is a mathematical program in which the objective function is linear in the unknowns and the constraint set is defined by a linear matrix inequality. This problem is nonlinear, nondifferentiable, but convex. It covers several standard problems (such as linear and quadratic programming) and has many applications in engineering. Typically, the optimal eigenvalue multiplicity associated with a linear matrix inequality is larger than one. Algorithms based on prior knowledge of the optimal eigenvalue multiplicity for solving the underlying problem have been shown to be efficient. In this paper, we propose a scheme to estimate the optimal eigenvalue multiplicity from points close to the solution. With some mild assumptions, it is shown that there exists an open neighborhood around the minimizer so that our scheme applied to any point in the neighborhood will always give the correct optimal eigenvalue multiplicity. We then show how to incorporate this result into a generalization of an existing local method for solving the semidefinite programming problem. Finally, a numerical example is included to illustrate the results.     

Global optimization for the generalized polynomial sum of ratios problem

In this paper, a new deterministic global optimization algorithm is proposed for solving a fractional programming problem whose objective and constraint functions are all defined as the sum of generalized polynomial ratios, which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. The proposed algorithm is based on reformulating the problem as a monotonic optimization problem, and it turns out that the optimal solution which is provided by the algorithm is adequately guaranteed to be feasible and to be close to the actual optimal solution. Convergence of the algorithm is shown and numerical examples are given to illustrate the feasibility and efficiency of the present algorithm.     

A numerical method for solving optimal control problems using state parametrization

A numerical method for solving a special class of optimal control problems is given. The solution is based on state parametrization as a polynomial with unknown coefficients. This converts the problem to a non-linear optimization problem. To facilitate the computation of optimal coefficients, an improved iterative method is suggested. Convergence of this iterative method and its implementation for numerical examples are also given.