Definition of the boundary in the local Charzynski-Tammi conjecture

According to the Charzynski-Tammi conjecture, the symmetrized Pick function is extremal in the problem on the estimate for the nth Taylor coefficient in the class of holomorphic univalent functions close to the identical one. In this paper we find the exact value of M ₄ such that the symmetrized Pick function is locally extremal in the problem on the estimate for the 4th Taylor coefficient in the class of holomorphic normalized univalent functions, whose module is bounded byM ₄.     

Maximization of the lift/drag ratio of airfoils with a turbulent boundary layer: Sharp estimates,approximation, and numerical solutions

The lift/drag ratio of an airfoil placed in an incompressible attached flow is maximized taking into account the viscosity in the boundary-layer approximation. An exact solution is constructed. The situation when the resulting solutions are not in the admissible class of univalent flows is discussed. A procedure is proposed for determining physically feasible airfoils (with a univalent flow region) with a high lift/drag ratio. For this purpose, a class of airfoils is constructed that are determined by a twoparameter function approximating the found exact solution to the variational problem. For this class, the ranges of free parameters leading to physically feasible flows are found. The results are verified by computing a turbulent boundary layer using Eppler’s method, and airfoils with a high lift/drag ratio in an attached flow are detected.     

Notes on Ruscheweyh's Multiplier Conjecture

In this note we show that Ruscheweyh's multiplier conjecture is true in several special cases. The results obtained here are closely related to the partial sums of certain analytic functions defined by means of univalent functions. We also answer a question of Ponnusamy concerning thenth partial sums of certain univalent functions.     

Generalizations of the area theorem for meromorphic univalent functions with nonzero pole

In this article, we consider meromorphic univalent functions f in the unit disc of the complex plane having a simple pole at z = α ∈ (0, 1) with nonzero residue b at z = α. In 1969, P.N. Chichra proved an area theorem for such functions. In this note, we generalize this theorem and prove an interesting consequence of this result.     

再论各向异性平面弹性中的一个周期接触问题——纪念李国平院士吴新谋教授诞辰100周年

该文对下面的一个弹性接触问题进行了再讨论.一列周期排列刚性压头压入一个各向异性弹性半平面的边界上,并有摩擦力存在, 求应力平衡.此问题在文献[1, 2]中曾讨论过并求出其解.但解法似不完全确切.该文将它进行修正从而获得精确解的封闭形式.     

An Estimation of Exact Penalty for Infinite-Dimensional Inequality-Constrained Minimization Problems

We use the penalty approach in order to study inequality-constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we consider a large class of inequality-constrained minimization problems for which a constraint is a mapping with values in a normed ordered space. For this class of problems we introduce a new type of penalty functions, establish the exact penalty property and obtain an estimation of the exact penalty. Using this exact penalty property we obtain necessary and sufficient optimality conditions for the constrained minimization problems.     

On the exact Markov inequality for k-monotone polynomials in uniform and L 1-norms

We consider the classical extremal problem of estimating norms of higher order derivatives of algebraic polynomials when their norms are given. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov, while Bernstein found the exact constant in the Markov inequality for monotone polynomials. In this note we give Markov-type inequalities for higher order derivatives in the general class of k-monotone polynomials. In particular, in case of first derivative we find the exact solution of this extremal problem in both uniform and L ₁-norms. This exact solution is given in terms of the largest zeros of certain Jacobi polynomials.     

Applications of Livingston‐type inequalities to the generalized Zalcman functional

We obtain sharp estimates for a generalized Zalcman coefficient functional with a complex parameter for the Hurwitz class and the Noshiro–Warschawski class of univalent functions as well as for the closed convex hulls of the convex and starlike functions by using an inequality from [6]. In particular, we generalize an inequality proved by Ma for starlike functions and answer a question from his paper [17]. Finally, we prove an asymptotic version of the generalized Zalcman conjecture for univalent functions and discuss various related or equivalent statements which may shed further light on the problem.     

A rotation theorem in the class of bounded univalent functions

Let S_M, M > 1, be the class of functionsf(z) which are regular and univalent in the disk ¦z¦ < 1 and satisfy the conditionsf(0) = 0,f'(0) = 1, and ¦f(z)¦ < M. In the present note we will obtain an exact estimate for the argument of the derivative of a function of the class S_M.     

Estimates for inverse coefficients of univalent functions from integral means

The purpose of this note is to point out that sharp coefficient bounds for the inverses of univalent functions from certain families are fairly direct corollaries of results on integral means. As an example, in §1 the method is applied to the familiar schlicht classS. The resulting coefficient estimates for the inverses of functions inS were first obtained by K. Löwner. Following this prototype, in §2 we obtain corresponding results, which are new, for a classS(p) of meromorphic schlicht functions in |z|<1.     

A note on constraint aggregation and value functions for two-stage stochastic integer programs

We consider a class of two-stage stochastic integer programs and their equivalent reformulation that uses the integer programming value functions in both stages. One class of solution methods in the literature is based on the idea of pre-computing and storing exact value functions, and then exploiting this information within a global branch-and-bound framework. Such methods are known to be very sensitive to the magnitude of feasible right-hand side values. In this note we propose a simple constraint-aggregation based approach that potentially alleviates this limitation.     

Decompositions of functions and Dirichlet problems in the unit disc

In this paper, we establish a decomposition theorem for polyharmonic functions and consider its applications to some Dirichlet problems in the unit disc. By the decomposition, we get the unique solution of the Dirichlet problem for polyharmonic functions (PHD problem) and give a unified expression for a class of kernel functions associated with the solution in the case of the unit disc introduced by Begehr, Du and Wang. In addition, we also discuss some quasi-Dirichlet problems for homogeneous mixed-partial differential equations of higher order. It is worthy to note that the decomposition theorem in the present paper is a natural extension of the Goursat decomposition theorem for biharmonic functions.     

On the Parameter-Uniform Convergence of Exponential Spline Interpolation in the Presence of a Boundary Layer

The paper is concerned with the problem of generalized spline interpolation of functions having large-gradient regions. Splines of the class C², represented on each interval of the grid by the sum of a second-degree polynomial and a boundary layer function, are considered. The existence and uniqueness of the interpolation L-spline are proven, and asymptotically exact two-sided error estimates for the class of functions with an exponential boundary layer are obtained. It is established that the cubic and parabolic interpolation splines are limiting for the solution of the given problem. The results of numerical experiments are presented.     

On the exactness of a class of nondifferentiable penalty functions

In this paper, we consider a class of nondifferentiable penalty functions for the solution of nonlinear programming problems without convexity assumptions. Preliminarily, we introduce a notion of exactness which appears to be of relevance in connection with the solution of the constrained problem by means of unconstrained minimization methods. Then, we show that the class of penalty functions considered is exact, according to this notion. This research was partially supported by the National Research Program on “Modelli e Algoritmi per l'Ottimizzazione,” Ministero della Pubblica, Istruzione, Roma, Italy.     

The theorems of Loewner and Pick

The study of the exact interpolation of quadratic norms in vector spaces depends in an essential way on the theory of monotone matrix functions developed by Loewner in 1934 [4]. This theory, in its turn, depends on Loewner’s solution of a problem of interpolation by rational functions of a certain class. The discussion of this latter problem is necessarily complicated, and Loewner’s text does not lend itself to ready reference. It has therefore seemed worthwhile to recast a portion of Loewner’s results in a form more suited to the applications we have in view. Our work, however, is not wholly derivative; none of our theorems are explicitly stated by Loewner and our arguments, which are of a more geometric character, are essentially different. The knowledgeable reader will note that our hypotheses are slightly stronger than Loewner’s and that our results are therefore also stronger. For the applications which we have in mind, Theorem III is the most important result; the proof of this theorem depends on all of the previously developed theory.     

On some results of A. E. Livingston and coefficient problems for concave univalent functions

We consider functions that map the open unit disc conformally onto the complement of a bounded convex set. We call these functions concave univalent functions. In 1994, Livingston presented a characterization for these functions. In this paper, we observe that there is a minor flaw with this characterization. We obtain certain sharp estimates and the exact set of variability involving Laurent and Taylor coefficients for concave functions. We also present the exact set of variability of the linear combination of certain successive Taylor coefficients of concave functions.     

一类新的罚函数与罚算法(英)

在本文中,我们提出了带不等式约束的非线性规划问题的一类新的罚函数,它的一个子类可以光滑逼近$l_1$罚函数. 基于此类新的罚函数我们给出了一种罚算法,这个算法的特点是每次迭代求出罚函数的全局精确解或非精确解. 在很弱的条件下算法总是可行的. 我们在不需要任何约束规范的情况下,证明了算法的全局收敛性. 最后给出了数值实验.     

SOS-Convex Semialgebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization

In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with ? ₁-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semialgebraic programs can be found by solving a single semidefinite programming problem (SDP). We achieve the results by using tools from semialgebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we show that robust SOS-convex optimization proble ms under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers an open question in the literature on how to recover a robust solution of uncertain SOS-convex polynomial programs from its semidefinite programming relaxation in this broader setting.     

A parallel clustering technique for the vehicle routing problem with split deliveries and pickups

In this paper, a parallel clustering technique and route construction heuristic have been developed for the vehicle routing problem (VRP) with split deliveries and pickups. An MILP formulation for determining the exact solution to the problem has also been included. It has been shown through extensive experimentation that the algorithm proposed in this paper statistically produces better results than the only heuristic existing for this class of problems in literature. We also form a basis of comparison between this class of problems and the VRP with simultaneous deliveries and pickups. We note that while heuristics for simultaneous deliveries and pickups cannot be applied in situations where customers' delivery or pickup demands exceed the vehicle capacity, heuristics allowing split deliveries and pickups can, in fact, be applied in every situation, even producing superior results under the combined objective of minimization of the fixed charge and mileage associated with vehicle routes. A guideline as to which heuristic could be used under what parametric conditions and objective functions, has also been provided.     

A Class of penalty functions for optimization problema with bound constraints

In this paper we propose a new class of continuously differentiable globally exact penalty functions for the solution of minimization problems with simple bounds on some (all) of the variables. The penalty functions in this class fully exploit the structure of the problem and are easily computable. Furthermore we introduce a simple updating rule for the penalty parameter that can be used in conjunction with unconstrained minimization techniques to solve the original problem.