A globally convergent version of the Polak-Ribière conjugate gradient method

In this paper we propose a new line search algorithm that ensures global convergence of the Polak-Ribière conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions. In particular, we show that with this line search every limit point produced by the Polak-Ribière iteration is a stationary point of the objective function. Moreover, we define adaptive rules for the choice of the parameters in a way that the first stationary point along a search direction can be eventually accepted when the algorithm is converging to a minimum point with positive definite Hessian matrix. Under strong convexity assumptions, the known global convergence results can be reobtained as a special case. From a computational point of view, we may expect that an algorithm incorporating the step-size acceptance rules proposed here will retain the same good features of the Polak-Ribière method, while avoiding pathological situations. This research was supported by Agenzia Spaziale Italiana, Rome, Italy.     

Three-term conjugate gradient method for the convex optimization problem over the fixed point set of a nonexpansive mapping

Many constrained sets in problems such as signal processing and optimal control can be represented as a fixed point set of a certain nonexpansive mapping, and a number of iterative algorithms have been presented for solving a convex optimization problem over a fixed point set. This paper presents a novel gradient method with a three-term conjugate gradient direction that is used to accelerate conjugate gradient methods for solving unconstrained optimization problems. It is guaranteed that the algorithm strongly converges to the solution to the problem under the standard assumptions. Numerical comparisons with the existing gradient methods demonstrate the effectiveness and fast convergence of this algorithm.     

On strong and total Lagrange duality for convex optimization problems

We give some necessary and sufficient conditions which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. We also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals. In the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so-called Farkas-Minkowski and locally Farkas-Minkowski conditions for systems of convex inequalities, recently used in the literature. Moreover, we show that our new results extend some existing ones in the literature.     

二维椭圆型方程反问题中优化算法的比较

近年来，决定椭圆型方程系数反问题在地磁、地球物理、冶金和生物等实际问题上有着广泛的应用．本文讨论了二维的决定椭圆型方程系数反问题的数值求解方法．由误差平方和最小原则，这个反问题可化为一个变分问题，并进一步离散化为一个最优化问题，其目标函数依赖于要决定的方程系数．本文着重考察非线性共轭梯度法在此最优化问题数值计算中的表现，并与拟牛顿法作为对比．为了提高算法的效率我们适当选择加快收敛速度的预处理矩阵．同时还考察了线搜索方法的不同对优化算法的影响．数值实验的结果表明，非线性共轭梯度法在这类大规模优化问题中相对于拟牛顿法更有效．     

Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.     

金属介质复合结构目标电磁散射特性

基于体-面混合积分方程方法,研究了介质导体复合结构目标的电磁散射特性.对导体采用面积分方程,对介质应用体积分方程,由等效原理建立以导体面电流及任意非均匀介质体极化电流为未知量的矩阵方程,利用稳定的双共轭梯度并结合快速傅立叶变换技术(BiCGS-FFT)来加速矩阵方程求解,相对于传统的矩量法大大降低了计算时间和内存需求.数值结果验证了该方法的正确性.同时,它为复杂复合结构目标散射特性研究奠定了一定的基础.     

Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization

We present the convergence analysis of the inexact infeasible path-following (IIPF) interior-point algorithm. In this algorithm, the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix. The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of the interior-point method for this specific inexact case. We present the convergence analysis of the inexact infeasible path-following (IIPF) algorithm, prove the global convergence of this method and provide complexity analysis. Communicated by Y. Zhang.     

An algorithm for maximizing entropy subject to simple bounds

An unusual form of the maximum entropy problem is considered, that includes simple bound constraints on the Fourier coefficients of the required image, as well as nonnegativity conditions on the image intensities. The algorithm avoids mixing these constraints by introducing a parameter into the objective function that is adjusted by an outer iteration. For each parameter value an inner iteration solves a large optimization calculation, whose constraints are just the simple bounds, by a combination of the conjugate gradient procedure and an active set method. An important feature is the ability to make many changes to the active set at once. The outer iteration includes a test for inconsistency of all the given constraints. The algorithm is described, a proof of convergence is given, and there are some second-hand remarks on numerical results.     

Commutators, C0-semigroups and resolvent estimates

We study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self-adjoint operator H in the framework of the conjugate operator method initiated by Mourre. We allow the conjugate operator A to be the generator of a C₀-semigroup (finer estimates require A to be maximal symmetric) and we consider situations where the first commutator [H,iA] is not comparable to H. The applications include the spectral theory of zero mass quantum field models.     

Scheduling of power generation via large-scale nonlinear optimization

We investigate methods for solving high-dimensional nonlinear optimization problems which typically occur in the daily scheduling of electricity production: problems with a nonlinear, separable cost function, lower and upper bounds on the variables, and an equality constraint to satisfy the demand. If the cost function is quadratic, we use a modified Lagrange multiplier technique. For a nonquadratic cost function (a penalty function combining the original cost function and certain fuel constraints, so that it is generally not separable), we compare the performance of the gradient-projection method and the reduced-gradient method, both with conjugate search directions within facets of the feasible set. Numerical examples at the end of the paper demonstrate the effectiveness of the gradient-projection method to solve problems with hundreds of variables by exploitation of the special structure.