一类不可微二次规划逆问题

本文求解了一类二次规划的逆问题,具体为目标函数是矩阵谱范数与向量无穷范数之和的最小化问题.首先将该问题转化为目标函数可分离变量的凸优化问题,提出用G-ADMM法求解.并结合奇异值阈值算法,Moreau-Yosida正则化算法,matlab优化工具箱的quadprog函数来精确求解相应的子问题.而对于其中一个子问题的精确求解过程中发现其仍是目标函数可分离变量的凸优化问题,由于其变量都是矩阵,所以采用适合多个矩阵变量的交替方向法求解,通过引入新的变量,使其每个子问题的解都具有显示表达式.最后给出采用的G-ADMM法求解本文问题的数值实验.数据表明,本文所采用的方法能够高效快速地解决该二次规划逆问题.     

Optimal consumption and portfolio selection with quadratic utility and a subsistence consumption constraint

In this article, we analyze the optimal consumption and investment policy of an agent who has a quadratic felicity function and faces a subsistence consumption constraint. The agent's optimal investment in the risky asset increases linearly for low wealth levels. Risk taking continues to increase at a decreasing rate for wealth levels higher than subsistence wealth until it hits a maximum at a certain wealth level, and declines for wealth levels above this threshold. Further, the agent has a bliss level of consumption, since if an agent consumes more than this level she will suffer utility loss. Eventually her risk taking becomes zero at a wealth level which supports her bliss consumption.     

Least-norm linear programming solution as an unconstrained minimization problem

It is shown that the dual of the problem of minimizing the 2-norm of the primal and dual optimal variables and slacks of a linear program can be transformed into an unconstrained minimization of a convex parameter-free globally differentiable piecewise quadratic function with a Lipschitz continuous gradient. If the slacks are not included in the norm minimization, one obtains a minimization problem with a convex parameter-free quadratic objective function subject to nonnegativity constraints only.     

Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L ² norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.     

凸规划的新算法

对一般凸目标函数和一般凸集约束的凸规划问题新解法进行探讨,它是线性规划一种新算法的扩展和改进,此算法的基本思想是在规划问题的可行域中由所建－的一个切割面到另一个切割面的不断推进来求取最优的。文章对目标函数是二次的且约束是一般凸集和二次目标函数且约束是线性的情形,给出了更简单的算法。     

An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints

In this paper, we propose a branch-and-bound algorithm for finding a global optimal solution for a nonconvex quadratic program with convex quadratic constraints (NQPCQC). We first reformulate NQPCQC by adding some nonconvex quadratic constraints induced by eigenvectors of negative eigenvalues associated with the nonconvex quadratic objective function to Shor’s semidefinite relaxation. Under the assumption of having a bounded feasible domain, these nonconvex quadratic constraints can be further relaxed into linear ones to form a special semidefinite programming relaxation. Then an efficient branch-and-bound algorithm branching along the eigendirections of negative eigenvalues is designed. The theoretic convergence property and the worst-case complexity of the proposed algorithm are proved. Numerical experiments are conducted on several types of quadratic programs to show the efficiency of the proposed method.     

A new piecewise quadratic approximation approach for L0 norm minimization problem

In this paper, we consider the problem of finding sparse solutions for underdetermined systems of linear equations, which can be formulated as a class of L_0 norm minimization problem. By using the least absolute residual approximation, we propose a new piecewis, quadratic function to approximate the L_0 norm.Then, we develop a piecewise quadratic approximation(PQA) model where the objective function is given by the summation of a smooth non-convex component and a non-smooth convex component. To solve the(PQA) model,we present an algorithm based on the idea of the iterative thresholding algorithm and derive the convergence and the convergence rate. Finally, we carry out a series of numerical experiments to demonstrate the performance of the proposed algorithm for(PQA). We also conduct a phase diagram analysis to further show the superiority of(PQA) over L_1 and L_(1/2) regularizations.     

Quadratic optimization problems in robust beamforming

This paper presents a class of constrained optimization problems whereby a quadratic cost function is to be minimized with respect to a weight vector subject to an inequality quadratic constraint on the weight vector. This class of constrained optimization problems arises as a result of a motivation for designing robust antenna array processors in the field of adaptive array processing. The constrained optimization problem is first solved by using the primal-dual method. Numerical techniques are presented to reduce the computational complexity of determining the optimal Lagrange multiplier and hence the optimal weight vector. Subsequently, a set of linear constraints or at most linear plus norm constraints are developed for approximating the performance achievable with the quadratic constraint. The use of linear constraints is very attractive, since they reduce the computational burden required to determine the optimal weight vector.     

线性等式约束多目标规划的一个降维算法

本文提出具有线性等式约束多目标规划问题的一个降维算法．当目标函数全是二次或线性但至少有一个二次型时，用线性加权法转化原问题为单目标二次规划，再用降维方法转化为求解一个线性方程组．若目标函数非上述情形，首先用线性加权法将原问题转化为具有线性等式约束的非线性规划，然后，对这一非线性规划的目标函数二次逼近，构成线性等式约束二次规划序列，用降维法求解，直到满足精度要求为止．     

On trust region methods for unconstrained minimization without derivatives

We consider some algorithms for unconstrained minimization without derivatives that form linear or quadratic models by interpolation to values of the objective function. Then a new vector of variables is calculated by minimizing the current model within a trust region. Techniques are described for adjusting the trust region radius, and for choosing positions of the interpolation points that maintain not only nonsingularity of the interpolation equations but also the adequacy of the model. Particular attention is given to quadratic models with diagonal second derivative matrices, because numerical experiments show that they are often more efficient than full quadratic models for general objective functions. Finally, some recent research on the updating of full quadratic models is described briefly, using fewer interpolation equations than before. The resultant freedom is taken up by minimizing the Frobenius norm of the change to the second derivative matrix of the model. A preliminary version of this method provides some very promising numerical results. Presented at NTOC 2001, Kyoto, Japan.     

Bilinear programming: An exact algorithm

The Bilinear Programming Problem is a structured quadratic programming problem whose objective function is, in general, neither convex nor concave. Making use of the formal linearity of a dual formulation of the problem, we give a necessary and sufficient condition for optimality, and an algorithm to find an optimal solution.Research partially supported by the Office of Naval Research under Contract N00014-69-A-0200-1010 with the University of California.     

Continuity of the optimal value function and optimal solutions of parametric mixed-integer quadratic programs

To properly describe and solve complex decision problems, research on theoretical properties and solution of mixed-integer quadratic programs is becoming very important. We establish in this paper different Lipschitz-type continuity results about the optimal value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of the linear constraints. The obtained results extend some existing results for continuous quadratic programs, and, more importantly, lay the foundation for further theoretical study and corresponding algorithm analysis on mixed-integer quadratic programs.     

Ranking in quadratic integer programming problems

The present paper develops an algorithm for ranking the integer feasible solutions of a quadratic integer programming (QIP) problem. A linear integer programming (LIP) problem is constructed which provides bounds on the values of the objective function of the quadratic problem. The integer feasible solutions of this related integer linear programming problem are systematically scanned to rank the integer feasible solutions of the quadratic problem in non-decreasing order of the objective function values. The ranking in the QIP problem is useful in solving a nonlinear integer programming problem in which some other complicated nonlinear restrictions are imposed which cannot be included in the simple linear constraints of QIP, the objective function being still quadratic.     

一类连续可分离背包问题的直接算法

对于一类带有单个线性约束以及盒约束的一般连续可分离二次背包问题给出了一种直接的算法,根据模型特有的结构,通过调节线性约束的拉格朗日乘子λ 的取值范围,以及在算法求解过程中通过判断目标函数一次项中的变量是否在盒约束范围内,来逐步确定所有变量的最优值, 并通过该算法得到的实验结果与其他算法的比较,说明了这种算法的可行性和有效性．     

On an inverse linear programming problem

A method for solving the following inverse linear programming (LP) problem is proposed. For a given LP problem and one of its feasible vectors, it is required to adjust the objective function vector as little as possible so that the given vector becomes optimal. The closeness of vectors is estimated by means of the Euclidean vector norm. The inverse LP problem is reduced to a problem of unconstrained minimization for a convex piecewise quadratic function. This minimization problem is solved by means of the generalized Newton method.     

Quadratic optimal synchronization of uncertain chaotic systems

This paper investigates the quadratic optimal synchronization of uncertain chaotic systems with parameter mismatch, parametric perturbations and external disturbances on both master and slave systems. A robust control scheme based on Lyapunov stability theory and quadratic optimal control approach is derived to realize chaotic synchronization. The sufficient criterion for stability condition is formulated in a linear matrix inequality (LMI) form. The effect of uncertain parameters and external disturbance is suppressed to an H_∞ norm constraint. An adaptive algorithm is proposed to adjust the uncertain bound in the robust controller avoiding the chattering phenomena. The simulation results for synchronization of the Chua’s circuit system and the Lorenz system demonstrate the effectiveness of the proposed scheme.     

一种非增值型凸二次双层规划的有效算法

主要研究了非增值型凸二次双层规划的一种有效求解算法。首先利用数学规划的对偶理论,将所求双层规划转化为一个下层只有一个无约束凸二次子规划的双层规划问题.然后根据两个双层规划的最优解和最优目标值之间的关系,提出一种简单有效的算法来解决非增值型凸二次双层规划问题.并通过数值算例的计算结果说明了该算法的可行性和有效性。     

Singular Arcs in the Generalized Goddard’s Problem

We investigate variants of Goddard’s problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle the problem of nonsmoothness of the optimal control. Support from the French Space Agency CNES (Centre National d’Etudes Spatial) is gratefully acknowledged.     

A new reduced gradient method for solving linearly constrained multiobjective optimization problems

In this paper, we consider the linearly constrained multiobjective minimization, and we propose a new reduced gradient method for solving this problem. Our approach solves iteratively a convex quadratic optimization subproblem to calculate a suitable descent direction for all the objective functions, and then use a bisection algorithm to find an optimal stepsize along this direction. We prove, under natural assumptions, that the proposed algorithm is well-defined and converges globally to Pareto critical points of the problem. Finally, this algorithm is implemented in the MATLAB environment and comparative results of numerical experiments are reported.     

A Line-Search Algorithm Inspired by the Adaptive Cubic Regularization Framework and Complexity Analysis

Adaptive regularized framework using cubics has emerged as an alternative to line-search and trust-region algorithms for smooth nonconvex optimization, with an optimal complexity among second-order methods. In this paper, we propose and analyze the use of an iteration dependent scaled norm in the adaptive regularized framework using cubics. Within such a scaled norm, the obtained method behaves as a line-search algorithm along the quasi-Newton direction with a special backtracking strategy. Under appropriate assumptions, the new algorithm enjoys the same convergence and complexity properties as adaptive regularized algorithm using cubics. The complexity for finding an approximate first-order stationary point can be improved to be optimal whenever a second-order version of the proposed algorithm is regarded. In a similar way, using the same scaled norm to define the trust-region neighborhood, we show that the trust-region algorithm behaves as a line-search algorithm. The good potential of the obtained algorithms is shown on a set of large-scale optimization problems.