等式约束优化的组合信赖域与拟牛顿算法

本文对等式约束问题提出了一个种组合信赖域与拟牛顿算法。该算法的特点是若Lagrangian函数的近似Ｈessian阵在等式约束Jacobi阵的零空间正定的,则选择拟牛顿算法,否则用信赖域算法,在通常信赖域算法的收敛假设下,该文证明了组合算法的全局收敛性。     

SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.     

Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient for global or local optimality under some MPEC generalized convexity assumptions. Moreover, we propose new constraint qualifications for M-stationary conditions to hold. These new constraint qualifications include piecewise MFCQ, piecewise Slater condition, MPEC weak reverse convex constraint qualification, MPEC Arrow-Hurwicz-Uzawa constraint qualification, MPEC Zangwill constraint qualification, MPEC Kuhn-Tucker constraint qualification, and MPEC Abadie constraint qualification.     

对称迭层矩形板的平面应力分析

常用的对称迭层板为各向异性板．根据平面应力问题的基本方程精确地用应力函数解法求得了各向异性板的一般解析解．推导出平面内应力和位移的一般公式,其中积分常数由边界条件来决定．一般解包括三角函数和双曲函数组成的解,它能满足4个边为任意边界条件的问题．还有代数多项式解,它能满足4个角的边界条件．因此一般解可用以求解任意边界条件下的平面应力问题．以4边承受均匀法向和切向载荷以及非均匀法向载荷的对称迭层方板为例,进行了计算和分析．     

Computational method for optimal control of switched systems with input and state constraints

In this paper, we consider an optimal control problem of switched systems with input and state constraints. Since the complexity of such constraint and switching laws, it is difficult to solve the problem using standard optimization techniques. In addition, although conjugate gradient algorithms are very useful for solving nonlinear optimization problem, in practical implementations, the existing Wolfe condition may never be satisfied due to the existence of numerical errors. And the mode insertion technique only leads to suboptimal solutions, due to only certain mode insertions being considered. Thus, based on an improved conjugate gradient algorithm and a discrete filled function method, an improved bi-level algorithm is proposed to solve this optimization problem. Convergence results indicate that the proposed algorithm is globally convergent. Three numerical examples are solved to illustrate the proposed algorithm converges faster and yields a better cost function value than existing bi-level algorithms.     

A (2+ϵ)-approximation for precedence constrained single machine scheduling with release dates and total weighted completion time objective

We give a $(2+?)$-approximation algorithm for minimizing total weighted completion time on a single machine under release time and precedence constraints. This settles a recent conjecture on the approximability of this scheduling problem (Skutella, 2016).     

Theoretical study on the mechanism of thieno[3,2‐b]benzofuran bromination: the importance of Lewis and non‐Lewis type NBOs interactions along the reaction path

A density functional theory study has been performed to estimate the electrophilic thieno[3,2‐b]benzofuran bromination reaction. Optimized structures for all stationary points were examined by employing the B3LYP and BMK at the 6‐31++G(d,p), 6‐311G(d,p), and 6‐311++G(d,p) levels of theory. The solvent polarity has a significant effect on a reduction of activation energies barriers. The reaction involves the formation of a triangle complex, migration of a proton through the bromine moiety followed by ionization of the bromine bond, and activation to the σ‐complex. Finally, the σ‐complex transforms into the reaction products. The natural bond orbital (NBO) population analysis was performed along the reaction minimal energy path defined as a function of the intrinsic reaction coordinate (IRC). The evolution of interaction energies between filled and empty NBOs along IRC has been estimated. The importance of these interactions for the disruption of Br?Br and C?H bonds and creation of C?Br and H?Br bonds have been emphasized. The changes in NBOs hybridization, covalency effects, electrostatic potential density maps, and occupancy of natural bonds have been investigated along IRC. The results obtained explain well the essence of bonding transformations and electron density changes during the reaction. Copyright © 2015 John Wiley & Sons, Ltd.     

Requiem for the Miller–Tucker–Zemlin subtour elimination constraints?

The Miller–Tucker–Zemlin (MTZ) Subtour Elimination Constraints (SECs) and the improved version by Desrochers and Laporte (DL) have been and are still in regular use to model a variety of routing problems. This paper presents a systematic way of deriving inequalities that are more complicated than the MTZ and DL inequalities and that, in a certain way, “generalize” the underlying idea of the original inequalities. We present a polyhedral approach that studies and analyses the convex hull of feasible sets for small dimensions. This approach allows us to generate generalizations of the MTZ and DL inequalities, which are “good” in the sense that they define facets of these small polyhedra. It is well known that DL inequalities imply a subset of Dantzig–Fulkerson–Johnson (DFJ) SECs for two-node subsets. Through the approach presented, we describe a generalization of these inequalities which imply DFJ SECs for three-node subsets and show that generalizations for larger subsets are unlikely to exist. Our study presents a similar analysis with generalizations of MTZ inequalities and their relation with the lifted circuit inequalities for three node subsets.     

弓张式超磁致伸缩材料换能器的设计与分析

为了提高超磁致伸缩换能器的作动行程,以获得足够大的发音强度,并使之满足发音装置体积小、装配零件少的要求,结合超磁致伸缩材料（GMM）的优良特性,提出一种基于三角放大原理的弓张式GMM换能器。该换能器以GMM棒作为驱动元件,通过固定弓张结构的一端,将双向输出转变为单向输出,同时利用柔性铰链结构,进一步增大换能器的位移输出。通过分析换能器的工作原理,计算得到其理论放大倍数为2.73,与所建立的有限元仿真模型计算得到的放大倍数2.8相近。制作了试验样机并搭建了相应的试验系统,得到在1 kHz范围内换能器最大输出位移为15.5 m,与仿真结果14.058 m相近。提出的弓张结构实现了换能器的位移放大,相应的分析方法也较好地反映了换能器的输出特性。     

A Class of Quadratic Programs with Linear Complementarity Constraints

We consider a class of quadratic programs with linear complementarity constraints (QPLCC) which belong to mathematical programs with equilibrium constraints (MPEC). We investigate various stationary conditions and present new and strong necessary and sufficient conditions for global and local optimality. Furthermore, we propose a Newton-like method to find an M-stationary point in finite steps without MEPC linear independence constraint qualification. The research of this author is partially supported by NSERC, and Research Grand Council of Hong Kong.