Mostafa Shamsi | AmirKabir University Of Technology (original) (raw)

Papers by Mostafa Shamsi

Mathematical Methods in the Applied Sciences, 2015

This paper presents a computational technique based on the pseudo-spectral method for the solutio... more This paper presents a computational technique based on the pseudo-spectral method for the solution of distributed optimal control problem for the viscous Burgers equation. By using pseudo-spectral method, the problem is converted to a classical optimal control problem governed by a system of ordinary differential equations, which can be solved by welldeveloped direct or indirect methods. For solving the resulting optimal control problem, we present an indirect method by deriving and numerically solving the first-order optimality conditions. Numerical tests involving both unconstrained and constrained control problems are considered.

In this paper, we consider the Homicidal Chauffeur (HC) problem as an interesting and practical d... more In this paper, we consider the Homicidal Chauffeur (HC) problem as an interesting and practical differential game. At first, we introduce a bilevel optimal control problem (BOCP) and prove that a saddle point solution for this game exists if and only if this BOCP has an optimal solution in which the optimal value of the objective function is equal to 1. Then, BOCP is discretized and converted to a nonlinear bilevel programming problem. Finally, an Artificial Bee Colony (ABC) algorithm is used for solving this problem, in which the lower-level problem will be considered as a constraint and solved by an NLP-solver. Finally, to demonstrate the effectiveness of the presented method, various cases of HC problem are solved and the simulation results are reported. Review History: Received:21 August 2019 Revised:12 October 2019 Accepted:15 October 2019 Available Online:01 September 2020

JMLR.orgPUB6573, Aug 1, 2010

Appropriate selection of the kernel function, which implicitly defines the feature space of an al... more Appropriate selection of the kernel function, which implicitly defines the feature space of an algorithm, has a crucial role in the success of kernel methods. In this paper, we consider the problem of optimizing a kernel function over the class of translation invariant kernels for the task of binary classification. The learning capacity of this class is invariant with respect to rotation and scaling of the features and it encompasses the set of radial kernels. We show that how translation invariant kernel functions can be embedded in a nested set of sub-classes and consider the kernel learning problem over one of these sub-classes. This allows the choice of an appropriate sub-class based on the problem at hand. We use the criterion proposed by Lanckriet et al. (2004) to obtain a functional formulation for the problem. It will be proven that the optimal kernel is a finite mixture of cosine functions. The kernel learning problem is then formulated as a semi-infinite programming (SIP) problem which is solved by a sequence of quadratically constrained quadratic programming (QCQP) sub-problems. Using the fact that the cosine kernel is of rank two, we propose a formulation of a QCQP sub-problem which does not require the kernel matrices to be loaded into memory, making the method applicable to large-scale problems. We also address the issue of including other classes of kernels, such as individual kernels and isotropic Gaussian kernels, in the learning process. Another interesting feature of the proposed method is that the optimal classifier has an expansion in terms of the number of cosine kernels, instead of support vectors, leading to a remarkable speedup at run-time. As a by-product, we also generalize the kernel trick to complex-valued kernel functions. Our experiments on artificial and real-world benchmark data sets, including the USPS and the MNIST digit recognition data sets, show the usefulness of the proposed method.

Optimization Methods and Software, 2022

Discrete & Continuous Dynamical Systems - B, 2019

We discuss and compare numerical methods to solve singular optimal control problems by the direct... more We discuss and compare numerical methods to solve singular optimal control problems by the direct method. Our discussion is illustrated by an Autonomous Underwater Vehicle (AUV) problem with state constraints. For this problem, we test four different approaches to solve numerically our problem via the direct method. After discretizing the optimal control problem we solve the resulting optimization problem with (i) A Mathematical Programming Language (AMPL), (ii) the Imperial College London Optimal Control Software (ICLOCS), (iii) the Gauss Pseudospectral Optimization Software (GPOPS) as well as with (iv) a new algorithm based on mixed-binary nonlinear programming reported in [7]. This algorithm consists on converting the optimal control problem to a Mixed Binary Optimal Control (MBOC) problem which is then transcribed to a mixed binary non-linear programming problem (MBNLP) problem using Legendre-Radau pseudospectral method. Our case study shows that, in contrast with the first three approaches we test (all relying on IPOPT or other numerical optimization software packages like KNITRO), the MBOC approach detects the structure of the AUV's problem without a priori information of optimal control and computes the switching times accurately.

Mathematical Methods in the Applied Sciences, 2017

In this paper, 2 extragradient methods for solving differential variational inequality (DVI) prob... more In this paper, 2 extragradient methods for solving differential variational inequality (DVI) problems are presented, and the convergence conditions are derived. It is shown that the presented extragradient methods have weaker convergence conditions in comparison with the basic fixed-point algorithm for solving DVIs. Then the linear complementarity systems, as an important and practical special case of DVIs, are considered, and the convergence conditions of the presented extragradient methods are adapted for them. In addition, an upper bound for the Lipschitz constant of linear complementarity systems is introduced. This upper bound can be used for adjusting the parameters of the extragradient methods, to accelerate the convergence speed. Finally, 4 illustrative examples are considered to support the theoretical results.

International Journal for Numerical Methods in Fluids, 2017

This paper presents a fast numerical method, based on the indirect shooting method and Proper Ort... more This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of Initial Value Problems (IVPs). Generally, these IVPs are high-order and thus their solution is time-consuming. To overcome this drawback, we present a POD indirect shooting method, which utilizes the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a Discrete Empirical Interpolation Method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of four numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas, there is no significant difference between the accuracy of the reduced and full indirect shooting methods. Copyright c

Applied Mathematical Modelling, 2015

In the present paper, an efficient pseudospectral method for the solution of two point boundary v... more In the present paper, an efficient pseudospectral method for the solution of two point boundary value problems arising in optimal control theory is presented. In the proposed method, the Gauss pseudospectral method is utilized to reduce a two point boundary value problem to the solution of a system of algebraic equations. However, convergence to the solution of the obtained system of equations may be slow or even fail, if a very good initial guess to the optimal solution is not available. To overcome this drawback, a numerical continuation method is used and the sensitivity of the proposed method to the initial guess is resolved. The main advantages of the present combined method are that, good results are obtained even by using a small number of discretization points and also the sensitivity to the initial guess for solving the final system of algebraic equations is resolved successfully. The presented method is useful, especially, where the shooting methods fail because of sensitivity or stiffness of the problem. Numerical results of two examples are presented at the end and efficiency of the combined method is reported.

arXiv: Optimization and Control, 2019

In this paper, we combine the positive aspects of the Gradient Sampling (GS) and bundle methods, ... more In this paper, we combine the positive aspects of the Gradient Sampling (GS) and bundle methods, as the most efficient methods in nonsmooth optimization, to develop a robust method for solving unconstrained nonsmooth convex optimization problems. The main aim of the proposed method is to take advantage of both GS and bundle methods, meanwhile avoiding their drawbacks. At each iteration of this method, to find an efficient descent direction, the GS technique is utilized for constructing a local polyhedral model for the objective function. If necessary, via an iterative improvement process, this initial polyhedral model is improved by some techniques inspired by the bundle and GS methods. The convergence of the method is studied, which reveals the following positive features (i) The convergence of our method is independent of the number of gradient evaluations required to establish and improve the initial polyhedral models. Thus, the presented method needs much fewer gradient evaluati...

International Journal of Control, 2017

This paper presents a new approach for the efficient and accurate solution of Singular Optimal Co... more This paper presents a new approach for the efficient and accurate solution of Singular Optimal Control Problems (SOCP). A novel feature of the proposed method is that it does not require a priori knowledge of the structure of the solution. As the first step of this method, the SOCP is converted into a binary optimal control problem. Then, by utilizing the pseudospectral method, the resulting problem is transcribed to a mixed-binary non-linear programming problem. This mixed-binary non-linear programming problem, which can be solved by well-known solvers, allows us to detect the structure of the original optimal control and to compute the approximating solution of it (getting both the optimal state and control). The main advantages of the present method are that: (i) without a priori information, the structure of optimal control is detected; (ii) it produces good results even using a small number of collocation points; and (iii) the switching times can be captured accurately. These advantages are illustrated through a numerical implementation of the method on four examples.

$Research paper thumbnail of Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems$

This paper presents three direct methods based on Grünwald-Letnikov, trapezoidal and Simpson frac... more This paper presents three direct methods based on Grünwald-Letnikov, trapezoidal and Simpson fractional integral formulas to solve fractional optimal control problems (FOCPs). At first, the fractional integral form of FOCP is considered, then the fractional integral is approximated by Grünwald-Letnikov, trapezoidal and Simpson formulas in a matrix approach. Thereafter, the performance index is approximated either by trapezoidal or Simpson quadrature. As a result, FOCP are reduced to nonlinear programming problems, which can be solved by many well-developed algorithms. To improve the efficiency of the presented method, the gradient of the objective function and the Jacobian of constraints are prepared in closed forms. It is pointed out that the implementation of the methods is simple and, due to the fact that there is no need to derive necessary conditions, the methods can be simply and quickly used to solve a wide class of FOCPs. The efficiency and reliability of the presented metho...

J. Mach. Learn. Res., 2010

A simple and effective method based on Haar wavelets is proposed for the solution of Pocklington&... more A simple and effective method based on Haar wavelets is proposed for the solution of Pocklington's integral equation. The properties of Haar wavelets are first given. These wavelets are utilized to reduce the solution of Pocklington's integral equation to the solution of algebraic equations. In order to save memory and computation time, we apply a thresho ld procedure to obtain sparse algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of resulted matrix equation.

Computational Optimization and Applications

In this paper, we combine the positive aspects of the gradient sampling (GS) and bundle methods, ... more In this paper, we combine the positive aspects of the gradient sampling (GS) and bundle methods, as the most efficient methods in nonsmooth optimization, to develop a robust method for solving unconstrained nonsmooth convex optimization problems. The main aim of the proposed method is to take advantage of both GS and bundle methods, meanwhile avoiding their drawbacks. At each iteration of this method, to find an efficient descent direction, the GS technique is utilized for constructing a local polyhedral model for the objective function. If necessary, via an iterative improvement process, this initial polyhedral model is improved by some techniques inspired by the bundle and GS methods. The convergence of the method is studied, which reveals that the global convergence property of our method is independent of the number of gradient evaluations required to establish and improve the initial polyhedral models. Thus, the presented method needs much fewer gradient evaluations in comparison to the original GS method. Furthermore, by means of numerical simulations, we show that the presented method provides promising results in comparison with GS methods, especially for large scale problems. Moreover, in contrast with some bundle methods, our method is not very sensitive to the accuracy of supplied gradients.

Engineering with Computers

In this paper, a fast computational technique based on adaptive mesh generation for numerical sol... more In this paper, a fast computational technique based on adaptive mesh generation for numerical solution of the obstacle problem is considered. The obstacle problem is an elliptic variational inequality problem, where its solution divides the domain into the contact and noncontact sets. The boundary between the contact and noncontact sets is a free boundary, which is priori unknown and the solution is not smooth on it. Due to lack of smoothness, numerical methods need a lot of mesh points in discretization for obtaining a numerical solution with a reasonable accuracy. In this paper, using an interpolating wavelet system and the fast wavelet transform, a multi-level algorithm for generating an appropriate-adapted mesh is presented. In each step of the algorithm, the semi-smooth Newton’s method or active set method is used for solving the discretized obstacle problem. We test the performance and accuracy of the proposed method by means of four numerical experiments. We show that the presented method significantly reduces CPU time in comparison with the full-grid algorithm and also can simultaneously capture the priori unknown free boundary.

Journal of Computational and Applied Mathematics

Abstract This paper presents an efficient numerical method for a class of Zero-Sum Pursuit-Evasio... more Abstract This paper presents an efficient numerical method for a class of Zero-Sum Pursuit-Evasion Differential Games (ZSPEDGs). The aim of the presented method is to resolve the drawbacks of the indirect methods in solving ZSPEDGs. In the indirect methods, the solution of ZSPEDG is found by solving a Two-Point Boundary Value Problem (TPBVP) derived from the necessary conditions. The indirect methods are accurate and fast. However, they are very sensitive to initial guess, and when the control is bounded, some non-smooth equations appear in the resulting TPBVP. These drawbacks restrict the use of indirect methods for solving ZSPEDGs. To overcome these drawbacks, at first, we reformulate the discontinuous equations of TPBVP by complementarity conditions, and as a result a Differential Complementarity System (DCS) is obtained. Then, the resulting DCS is reformulated to an optimal control problem, which can be solved by a well-developed direct or indirect method. The efficiency and robustness of the method are reported by means of two benchmarks and two real-life ZSPEDG problems.

Computers & Mathematics with Applications

Abstract This paper presents a fast computational technique based on the wavelet collocation meth... more Abstract This paper presents a fast computational technique based on the wavelet collocation method for the numerical solution of an optimal control problem governed by elliptic variational inequalities of obstacle type. In this problem, the solution divides the domain into contact and noncontact sets. The boundary between the contact and noncontact sets is a free boundary, which is not known a priori and the solution is not smooth on it. Accordingly, a very fine grid is needed in order to obtain a solution with a reasonable accuracy. In this paper, our aim is to propose an adaptive scheme in order to generate an appropriate and economic irregular dyadic mesh for finding the optimal control and state functions. The irregular mesh will be generated such that its density around the free boundary is higher than in other places and high-resolution computations are focused on these zones. To this aim, we use an adaptive wavelet collocation method and take advantage of the fast wavelet transform of compact-supported interpolating wavelets to develop a multi-level algorithm, which generates an adaptive computational grid. Using this adaptive grid takes less CPU time than using a full regular mesh. At each step of the algorithm, the active set method is used for solving the optimality system of the obstacle problem on the adapted mesh. Finally, the numerical examples are presented to show the validity and efficiency of the technique.

Ad Hoc Networks

The energy consumption is one of the most common issues in the Wireless Sensor Networks (WSNs). S... more The energy consumption is one of the most common issues in the Wireless Sensor Networks (WSNs). Since the communication usually accounts as a major power consumption, there is some techniques, such as topology control and network coding, to decrease the activity of sensors' transceivers. If we utilize the techniques synchronously, then may overall performance do not increase as expected. This paper provides an optimization problem for energy consumption in WSNs, where the network employs both topology control and network-codingbased multi-cast simultaneously. This approach improves overall performance in comparison with employing them distinctly. The proposed optimization problem is transformed into a convex problem which leads to a numerous theoretical and conceptual advantages. Then the Karush-Kuhn-Tucker (KKT) optimality conditions are presented to derive analytical expressions of the globally optimal solution. Simulation results show that the proposed approach decrease end-toend delay and has a significantly lower energy consumption than conventional ones.

Journal of Mathematical Analysis and Applications, 1991

This paper discusses the problem of finding the function U(X, t) and the unknown positive coeffic... more This paper discusses the problem of finding the function U(X, t) and the unknown positive coefficient a(t) in the parabolic initial-boundary value problem u,-u(t)du=O in 52 x (0, T] 44 t) =.0x, f) on ~22 x [0, T] (1.1) 24(x, 0) = h(x) in Q, where Q is a simply connected domain in R" with smooth boundary 852. With only the above data this problem is under-determined and we are forced to impose additional boundary conditions. We shall show that a unique solution pair (u, a) is obtained when in addition one prescribes certain time dependent functions of U. In particular, this may take the form of the heat flux g(t) at a given point x0 E X?, that is, 572