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Maximal information coefficient (MIC) is an indicator to explore the correlation between pairwise variables in large data sets, and the accuracy of MIC has an impact on the measure of dependence for each pair. To improve the equitability in an acceptable run-time, in this paper, an intelligent MIC (iMIC) is proposed for optimizing the partition on the y-axis to approximate the MIC with good accuracy. It is an iterative algorithm on quadratic optimization to generate a better characteristic matrix. During the process, the iMIC can quickly find out the local optimal value while using a lower number of iterations. It produces results that are close to the true MIC values by searching just n times, rather than n 2 computations required for the previous method. In the compared experiments of 169 indexes about 202 countries from World Health Organization (WHO) data set, the proposed algorithm offers a better solution coupled with a reasonable run-time for MIC, and good performance search for the extreme values in fewer iterations. The iMIC develops the equitability keeping the satisfied accuracy with fast computational speed, potentially benefitting the relationship exploration in big data. [ABSTRACT FROM AUTHOR]

In this paper, we try to study the numerical methods for solving integral equations from a new perspective—machine learning method. By means of the idea of kernel ε -support vector regression machine ( ε -SVR), we construct an optimization modeling for a class of Volterra–Fredholm integral equations and propose a novel numerical method for solving them. The proposed method has a certain versatility and can be used to solve some other kinds of integral equations. In order to verify the effectiveness of the proposed method, we perform a series of comparative experiments with six specific Volterra–Fredholm integral equations and a method proposed in Wang et al. (2014). Experimental results show that the proposed method has a good approximation property. [ABSTRACT FROM AUTHOR]

In probabilistic model analysis and optimization, expected values of a model output f ( x ) in face of continuous random inputs x are estimated through n -dimensional integrals, where n = d i m ( x ) . Cubature formulae are approximations of these integrals by a weighted sum of function evaluations at carefully chosen points. When each function evaluation corresponds to a heavy computational simulation, and particularly in optimization problems, one needs very efficient formulae with few integration points, even though only having modest accuracy. In this paper, we evaluate the performance of several cubature formulae with few points, including Smolyak type formulae, also known as sparse grid integration, and recently proposed thinned cubatures, constructed using orthogonal arrays. Tests are made for a wide family of smooth and non-oscillatory functions f ( x ) , possibly with significant anisotropy, and covering both normal and uniform input probability distributions. Two practical case studies are also presented, one of analysis of a large scale mass transfer model with uncertain parameters and a second one of optimal production planning under uncertain market conditions. Results clearly indicate that cubatures with large negative weights, including Smolyak type formulae, are not reliable, contrary to positive thinned cubatures that produce very reasonable estimates up to dimension 24. These thinned cubatures may also surpass quasi-Monte Carlo methods also up to dimension 24. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the low rank approximation of the symmetric positive semidefinite matrix, which arises in machine learning, quantum chemistry and inverse problem. We first characterize the feasible set by , and then transform low rank approximation into an unconstrained optimization problem. Finally, we use the nonlinear conjugate gradient method with exact line search to compute the optimal low rank symmetric positive semidefinite approximation of the given matrix. Numerical examples show that the new method is feasible and effective. [Copyright &y& Elsevier]

Abstract: In this paper we present a quasi-Newton’s method for unconstrained multiobjective optimization of strongly convex objective functions. Hence, we can approximate the Hessian matrices by using the well known BFGS method. The approximation of the Hessian matrices is usually faster than their exact evaluation, as used in, e.g., recently proposed Newton’s method for multiobjective optimization. We propose and analyze a new algorithm and prove that its convergence is superlinear. [Copyright &y& Elsevier]

Abstract: Approximation of rational B-spline curves by B-spline curves is an important issue in computer aided geometric design. This paper presents a method to approximate a uniform rational B-spline with B-spline curve sequence as follows. We first elevate the degree of the original rational B-spline curve and take the control points of the degree-elevated curve as new control points of the B-spline approximation curve. Next we take an extended knot vector of the original curve as a new knot vector of the approximation curve. This generates a B-spline approximation curve with the same degree as the degree-elevated curve. Based on the discrete B-spline and multiple products of B-spline functions, we finally prove that the derivatives of any given degree of the uniform B-spline approximation curve sequence converge uniformly to the corresponding derivatives of the original rational B-spline curve. This approximation method is very simple and guarantees the convergence of the approximation. [Copyright &y& Elsevier]

Abstract: In this paper we introduce several concepts of approximate solutions of set-valued optimization problems with vector and set optimization. We prove existence results and necessary and sufficient conditions by using limit sets. [Copyright &y& Elsevier]

Abstract: Expensive optimization aims to find the global minimum of a given function within a very limited number of function evaluations. It has drawn much attention in recent years. The present expensive optimization algorithms focus their attention on metamodeling techniques, and call existing global optimization algorithms as subroutines. So it is difficult for them to keep a good balance between model approximation and global search due to their two-part property. To overcome this difficulty, we try to embed a metamodel mechanism into an efficient evolutionary algorithm, low dimensional simplex evolution (LDSE), in this paper. The proposed algorithm is referred to as the low dimensional simplex evolution extension (LDSEE). It is inherently parallel and self-contained. This renders it very easy to use. Numerical results show that our proposed algorithm is a competitive alternative for expensive optimization problems. [Copyright &y& Elsevier]

Abstract: In this paper, a new approximation method is introduced to characterize a so-called vector strict global minimizer of order 2 for a class of nonlinear differentiable multiobjective programming problems with -convex functions of order 2. In this method, an equivalent vector optimization problem is constructed by a modification of both the objectives and the constraint functions in the original multiobjective programming problem at the given feasible point. In order to prove the equivalence between the original multiobjective programming problem and its associated -approximated vector optimization problem, the suitable -convexity of order 2 assumption is imposed on the functions constituting the considered vector optimization problem. [Copyright &y& Elsevier]

Abstract: Recently, Hager and Zhang (2005) proposed a new conjugate gradient method which generates sufficient descent direction , this property is independent of the line search used. In this paper, we take a modification of this method, such that the sufficient descent direction satisfies , this property is also independent of the line search used. Under appropriate conditions, we prove that the proposed method is globally convergent. Moreover, we give a sufficient condition for the global convergence of the proposed general method. The numerical results show that the proposed method is efficient. [Copyright &y& Elsevier]

Abstract: In this paper we discuss multiperiod portfolio selection problems related to a specific provisioning problem. Our results are an extension of Dhaene et al. (2005) , where optimal constant mix investment strategies are obtained in a provisioning and savings context, using an analytical approach based on the concept of comonotonicity. We derive convex bounds that can be used to estimate the provision to be set up at a specified time in future, to ensure that, after having paid all liabilities up to that moment, all liabilities from that moment on can be fulfilled, with a high probability. [Copyright &y& Elsevier]

Abstract: In the current paper, we study the convergence properties of the DGFE approximation of optimal control problem governed by convection–diffusion equations. We derive a posteriori error estimates and a priori error estimates for both the states, ad-joint and the control variable approximation. For the optimal control problem, these estimates are apparently not available in the literature. [Copyright &y& Elsevier]

Abstract: Recovering an unknown low-rank or approximately low-rank matrix from a sampling set of its entries is known as the matrix completion problem. In this paper, a nonlinear constrained quadratic program problem concerning the matrix completion is obtained. A new algorithm named the projected Landweber iteration (PLW) is proposed, and the convergence is proved strictly. Numerical results show that the proposed algorithm can be fast and efficient under suitable prior conditions of the unknown low-rank matrix. [ABSTRACT FROM AUTHOR]

Abstract: Modeling strategies often result in dynamical systems of very high dimension. It is then desirable to find systems of the same form but of lower complexity, whose input–output behavior approximates the behavior of the original system. Here we consider linear time-invariant discrete-time dynamical systems. The cornerstone of this paper is a relation between optimal model reduction in the -norm and (tangential) rational Hermite interpolation. First order necessary conditions for -optimal model reduction are presented for discrete Multiple-Input–Multiple-Output (MIMO) systems. These conditions suggest a specific choice of interpolation data and a novel algorithm aiming for an-optimal model reduction for MIMO systems. It is also shown that the conditions are equivalent to two known gramian-based first order necessary conditions. Numerical experiments demonstrate the approximation quality of the method. [Copyright &y& Elsevier]

Abstract: Functions with poles occur in many branches of applied mathematics which involve resonance phenomena. Such functions are challenging to interpolate, in particular in higher dimensions. In this paper we develop a technique for interpolation with quotients of two radial basis function (RBF) expansions to approximate such functions as an alternative to rational approximation. Since the quotient is not uniquely determined we introduce an additional constraint, the sum of the RBF-norms of the numerator and denominator squared should be minimal subjected to a norm condition on the function values. The method was designed for antenna design applications and we show by examples that the scattering matrix for a patch antenna as a function of some design parameters can be approximated accurately with the new method. In many cases, e.g. in antenna optimization, the function evaluations are time consuming, and therefore it is important to reduce the number of evaluations but still obtain a good approximation. A sensitivity analysis of the new interpolation technique is carried out and it gives indications how efficient adaptation methods could be devised. A family of such methods are evaluated on antenna data and the results show that much performance can be gained by choosing the right method. [Copyright &y& Elsevier]

Abstract: In this paper we propose a fundamentally different conjugate gradient method, in which the well-known parameter is computed by an approximation of the Hessian/vector product through finite differences. For search direction computation, the method uses a forward difference approximation to the Hessian/vector product in combination with a careful choice of the finite difference interval. For the step length computation we suggest an acceleration scheme able to improve the efficiency of the algorithm. Under common assumptions, the method is proved to be globally convergent. It is shown that for uniformly convex functions the convergence of the accelerated algorithm is still linear, but the reduction in function values is significantly improved. Numerical comparisons with conjugate gradient algorithms including CONMIN by Shanno and Phua [D.F. Shanno, K.H. Phua, Algorithm 500, minimization of unconstrained multivariate functions, ACM Trans. Math. Softw. 2 (1976) 87–94], SCALCG by Andrei [N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Comput. Optim. Appl. 38 (2007) 401–416; N. Andrei, Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Optim. Methods Softw. 22 (2007) 561–571; N. Andrei, A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Appl. Math. Lett. 20 (2007) 645–650], and new conjugacy condition and related new conjugate gradient by Li, Tang and Wei [G. Li, C. Tang, Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math. 202 (2007) 523–539] or truncated Newton TN by Nash [S.G. Nash, Preconditioning of truncated-Newton methods, SIAM J. on Scientific and Statistical Computing 6 (1985) 599–616] using a set of 750 unconstrained optimization test problems show that the suggested algorithm outperforms these conjugate gradient algorithms as well as TN. [Copyright &y& Elsevier]

Abstract: In this paper we develop a new class of conjugate gradient methods for unconstrained optimization problems. A new nonmonotone line search technique is proposed to guarantee the global convergence of these conjugate gradient methods under some mild conditions. In particular, Polak–Ribiére–Polyak and Liu–Storey conjugate gradient methods are special cases of the new class of conjugate gradient methods. By estimating the local Lipschitz constant of the derivative of objective functions, we can find an adequate step size and substantially decrease the function evaluations at each iteration. Numerical results show that these new conjugate gradient methods are effective in minimizing large-scale non-convex non-quadratic functions. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations. They come from two modified conjugate gradient methods [W.Y. Cheng, A two term PRP based descent Method, Numer. Funct. Anal. Optim. 28 (2007) 1217–1230; L. Zhang, W.J. Zhou, D.H. Li, A descent modified Polak–Ribiére–Polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal. 26 (2006) 629–640] recently proposed for unconstrained optimization problems. Under appropriate conditions, the global convergence of the proposed method is established. Preliminary numerical results show that the proposed method is promising. [Copyright &y& Elsevier]