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Solving Fractional Polynomial Problems by Polynomial Optimization Theory

Andrea Pizzo, Alessio Zappone, Luca Sanguinetti
2018 IEEE Signal Processing Letters  

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2018 pre-print arXiv:1806.07220v1 fatcat:my2njz6nj5gktoznrd7hhb6ydq
This work aims to introduce the framework of polynomial optimization theory to solve fractional polynomial problems (FPPs).  ...  Unlike other widely used optimization frameworks, the proposed one applies to a larger class of FPPs, not necessarily defined by concave and convex functions.  ...  Combined with the classical fractional programming theory [7] , we show how the polynomial optimization theory can be used to globally solve (1) as the order of the SOS reformulation grows to infinity  ... 
doi:10.1109/lsp.2018.2864620 fatcat:iqcyeg3qijfc7kk723rjgj6xby
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Andrea Pizzo, Alessio Zappone, Luca Sanguinetti. "Solving Fractional Polynomial Problems by Polynomial Optimization Theory." IEEE Signal Processing Letters 25.10 (2018) 1540-1544

Solving the optimal PWM problem for single-phase inverters

D. Czarkowski, D.V. Chudnovsky, I.W. Selesnick
2002 IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications  

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Moreover, it is shown that the polynomials ( ) associated with the optimal PWM problem are orthogonal and can therefore be obtained via simple recursions.  ...  Specifically, it is shown that the nonlinear design equations given by the standard mathematical formulation of the problem can be reformulated, and that the sought solution can be found by computing the  ...  For the general optimal PWM problem, the values can be obtained by solving the linear system (6). Example 1 (Harmonic Elimination): Let , .  ... 
doi:10.1109/81.995661 fatcat:zttjk6uawnax3imgmvzgh77d5u
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D. Czarkowski, D.V. Chudnovsky, I.W. Selesnick. "Solving the optimal PWM problem for single-phase inverters." IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications 49.4 (2002) 465-475

Review of Solution Methods for the Fractional Optimal Control

Sandip Anilbhai Mehta, Dipak Mukundray Adhyaru
2019 ICIC Express Letters  

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It covers the formulation of optimal control problem, approximation of fractional optimal control problem and solution methodologies for the fractional optimal control problem.  ...  Optimal control was introduced in 1954 by Pontryagin and Bellman. The fractional calculus of variation gains the momentum in last three decades.  ...  For example, in [37] the Shifted Legendre polynomial based operation matrix is formed to solve the fractional optimal control problem.  ... 
doi:10.24507/icicel.13.11.995 fatcat:icbxpsgdmnhulp5jiodt7cdvdm
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Sandip Anilbhai Mehta, Dipak Mukundray Adhyaru. "Review of Solution Methods for the Fractional Optimal Control." ICIC Express Letters 13.11 (2019) 995

Page 6520 of Mathematical Reviews Vol. , Issue 89K [page]

1989 Mathematical Reviews  

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The problem of constructing polynomials Z;(u) of step j —1 is solved by substituting the sequence of numbers Y\(A;),---, Yn41(A,) by interpolating polynomials of Lagrange Z;,\().  ...  The polynomial equation approach (the polynomial calculus) is used in solving various typical control problems in discrete-time and continuous-time linear systems.  ... 
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Mathematical Reviews (1989) 6520

Page 4497 of Mathematical Reviews Vol. , Issue 87h [page]

1987 Mathematical Reviews  

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Combinatorial optimization problems in class P (those that can be solved on a sequential machine in time polynomial in n, the size of the problem) and in class NP-complete (problems that require exponential  ...  optimization problems.  ... 
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Mathematical Reviews (1987) 4497

Quantum advantage for combinatorial optimization problems, Simplified [article]

Mario Szegedy
2022 arXiv   pre-print

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We observe that fault-tolerant quantum computers have an optimal advantage over classical computers in approximating solutions to many NP optimization problems.  ...  Similar theorems however immediately follow from the PCP theory without further ado. Let us take any NP optimization (NPO) problem Π.  ...  With this in mind, claims of solving select hard instances of an approximation problem with quantum is no more surprising than: "A simple quantum advantage for exactly solving combinatorial optimization  ... 
arXiv:2212.12572v1 fatcat:hqspzzfrp5bxxgk2zbok3slfwi
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Mario Szegedy. "Quantum advantage for combinatorial optimization problems, Simplified." arXiv (2022)

Approximation methods for solving fractional optimal control problems

Samaneh Soradi Zeid, Sohrab Effati, Ali Vahidian Kamyad
2017 Computational and Applied Mathematics  

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In this review paper, approximation methods for the free final time of fractional optimal control problems (FOCPs) are displayed.  ...  In this way, the considered tools and techniques mainly include the necessary optimal conditions in the form of two-point boundary value (TPBV) problem of fractional order.  ...  Fractional optimal control theory is a very new area in mathematics.  ... 
doi:10.1007/s40314-017-0424-2 fatcat:fxlzkpvljrgtrd6hxfj7m6idbm
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Samaneh Soradi Zeid, Sohrab Effati, Ali Vahidian Kamyad. "Approximation methods for solving fractional optimal control problems." Computational and Applied Mathematics 37.S1 (2017)

Solution of Some Types for Composition Fractional Order Differential Equations Corresponding to Optimal Control Problems

Sameer Qasim Hasan, Moataz Abbas Holel
2018 Journal of Control Science and Engineering  

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The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail.  ...  problems.  ...  In [6] , Khader and Hendy studied an efficient numerical scheme for solving fractional optimal control problems.  ... 
doi:10.1155/2018/3767263 fatcat:7tylm7hwo5bi7knmhbuakcrkxm
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Sameer Qasim Hasan, Moataz Abbas Holel. "Solution of Some Types for Composition Fractional Order Differential Equations Corresponding to Optimal Control Problems." Journal of Control Science and Engineering (2018) 1-12

Approximate solutions for a certain class of fractional optimal control problems using Laguerre collocation method

M. Adel, M. M. Khader
2017 International Mathematical Forum  

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The spectral Laguerre collocation method is presented for solving a class of fractional optimal control problems (FOCPs).  ...  The properties of Laguerre polynomials approximation and Rayleigh-Ritz method are used to reduce FOCPs to solve a system of algebraic equations which solved using Newton iteration method.  ...  Problem 1 (Linear time-invariant problem) Consider the following linear time invariant problem, which described by the following fractional optimal control problem ([1], [2] ) minimum J = 1 2 1 0 [x 2  ... 
doi:10.12988/imf.2017.7115 fatcat:3izbficedvbofmchqcgjzheuxy
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M. Adel, M. M. Khader. "Approximate solutions for a certain class of fractional optimal control problems using Laguerre collocation method." International Mathematical Forum (2017) 379-389

An Operational Matrix Method Based on the Gegenbauer Polynomials for Solving a Class of Fractional Optimal Control Problems

Farzaneh Soufivand, Fahimeh Soltanian, Kamal Mamehrashi
2021 International journal of industrial electronics, control and optimization  

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can be easily solved by Newton's iterative method.  ...  One of the most important classes of fractional calculus is the fractional optimal control problem (FOCP), which arises in engineering.  ...  Acknowledgment The authors are very grateful for all the constructive comments and suggestions by the reviewers to improve the paper.  ... 
doi:10.22111/ieco.2021.39546.1371 doaj:38fee9a017d44486873d02ddc710c8f9 fatcat:rfrl65eohzfd5nrhlmrxrgxrsm
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Farzaneh Soufivand, Fahimeh Soltanian, Kamal Mamehrashi. "An Operational Matrix Method Based on the Gegenbauer Polynomials for Solving a Class of Fractional Optimal Control Problems." International journal of industrial electronics, control and optimization (2021) 475-484

George Dantzig's impact on the theory of computation

Richard M. Karp
2008 Discrete Optimization  

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of the simplex algorithm and the intrinsic complexity of linear programming and combinatorial optimization.  ...  This paper traces a single historical thread: Dantzig's work on linear programming and its application and extension to combinatorial optimization, and the investigations it has stimulated about the performance  ...  his early work on cutting-plane methods, in which integer programming problems are solved by introducing linear constraints to eliminate fractional solutions.  ... 
doi:10.1016/j.disopt.2006.12.004 fatcat:ottc2vlxknbvxbzajpoidab7ri
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Richard M. Karp. "George Dantzig's impact on the theory of computation." Discrete Optimization 5.2 (2008) 174-185

Discrete Chebyshev Polynomials for Solving Fractional Variational Problems

Fakhrodin Mohammadi, Leila Moradi, Dajana Conte
2021 Statistics, Optimization and Information Computing  

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Then,‎ a numerical scheme based on the discrete Chebyshev polynomials and their operational matrix has been developed to solve fractional variational problems‎.  ...  Moreover, ‎the obtained numerical results ‎‎‎‎were compared to the previously acquired results by the classical Chebyshev polynomials.  ...  In the field of optimization theory, calculus of variations concerns with the problem of optimizing a realvalued functional over a set of functions [11, 12, 13, 14] .  ... 
doi:10.19139/soic-2310-5070-991 fatcat:j4wtn5emsbhw7nwm4262qcqc5y
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Fakhrodin Mohammadi, Leila Moradi, Dajana Conte. "Discrete Chebyshev Polynomials for Solving Fractional Variational Problems." Statistics, Optimization and Information Computing 9.3 (2021) 502-515

Generalization of dual structural optimization problems in terms of fractional programming

A. J. Morris
1978 Quarterly of Applied Mathematics  

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The paper makes a limited attempt to satisfy this requirement through a generalization of the associated duality theory by formulating the structural optimization as a fractional program.  ...  Duality theory has played an important role in the development of structural optimization theory and the associated computer-based solution algorithms.  ...  Since many structural optimization problems can be described in terms of homogeneous polynomials when cast in fractional form, this formulation is adhered to in the sequel.  ... 
doi:10.1090/qam/496698 fatcat:s6i27eurkvenbi5ujjzpfyyo54
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A. J. Morris. "Generalization of dual structural optimization problems in terms of fractional programming." Quarterly of Applied Mathematics 36.2 (1978) 115-119

Solving a class of fractional optimal control problems by the Hamilton–Jacobi–Bellman equation

Seyed Ali Rakhshan, Sohrab Effati, Ali Vahidian Kamyad
2016 Journal of Vibration and Control  

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This paper is dedicated to presenting an efficient numerical algorithm for solving a class of fractional optimal control problems (FOCPs).  ...  With the aid of the spectral-tau method, the problem can be reduced into a system of algebraic equations which can be solved via any suitable solver.  ...  FOCP reformulation The fractional optimal control problem (FOCP) can be defined as follows.  ... 
doi:10.1177/1077546316668467 fatcat:zzdzl7qzrvghbouiqinae4k7ee
Citation

Seyed Ali Rakhshan, Sohrab Effati, Ali Vahidian Kamyad. "Solving a class of fractional optimal control problems by the Hamilton–Jacobi–Bellman equation." Journal of Vibration and Control 24.9 (2016) 1741-1756

Optimal control of system governed by nonlinear volterra integral and fractional derivative equations

Leila Moradi, Dajana Conte, Eslam Farsimadan, Francesco Palmieri, Beatrice Paternoster
2021 Computational and Applied Mathemathics  

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A spectral approach is implemented based on the new polynomials known as Chelyshkov polynomials. First, the properties of these polynomials are studied to solve the aforementioned problems.  ...  AbstractThis work presents a novel formulation for the numerical solution of optimal control problems related to nonlinear Volterra fractional integral equations systems.  ...  To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.  ... 
doi:10.1007/s40314-021-01541-3 fatcat:nhlndfg2a5dv3ipb3ydfjxayfi
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Leila Moradi, Dajana Conte, Eslam Farsimadan, Francesco Palmieri, Beatrice Paternoster. "Optimal control of system governed by nonlinear volterra integral and fractional derivative equations." Computational and Applied Mathemathics 40.4 (2021)

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