Nevanlinna–Pick Problem and Uniqueness of Left Inverses in Convex Domains,Symmetrized Bidisc and Tetrablock

In the paper we discuss the problem of uniqueness of left inverses (solutions of two-point Nevanlinna–Pick problem) in bounded convex domains, strongly linearly convex domains, the symmetrized bidisc and the tetrablock.     

Linearly convex domains with singularities on the boundary

We present a topological classification of linearly convex domains with almost smooth boundary whose singularities lie in a hyperplane. We investigate sets with linearly convex boundary and the closures of linearly convex domains.     

On Locally Linearly Convex Domains

We construct a counterexample to the hypothesis on global linear convexity of locally linearly convex domains with everywhere smooth boundary. We refine the theorem on the topological classification of linearly convex domains with smooth boundary.     

On linearly convex domains with smooth boundaries

We establish that an arbitrary locally linearly convex domain with a smooth boundary is strongly linearly convex. Chernigov Pedagogic Institute, Chernigov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 11, pp. 1553–1556, November, 1997.     

有界凸平衡域上的双全纯凸映照的判别准则

本文讨论Cn中有界强凸平衡域和凸平衡域上局部双全纯映照成为双全纯凸映照的充要条件,从而得到Reinhardt域Dp= 上双全纯凸映照的充要条件,其中Pj≥2(j=1,2,…,n).     

Separation theorems for convex polytopes and finitely-generated cones derived from theorems of the alternative

We derive from Motzkin’s Theorem that a point can be strongly separated by a hyperplane from a convex polytope and a finitely-generated convex cone. We state a similar result for Tucker’s Theorem of the alternative. A generalisation of the residual existence theorem for linear equations which has recently been proved by Rohn [8] is a corollary. We state all the results in the setting of a general vector space over a linearly ordered (possibly skew) field.     

Klee Theorem for Linearly Convex Sets

We prove a complex analog of the classical Klee theorem for strongly linearly convex closed sets.     

Random Block Coordinate Descent Methods for Linearly Constrained Optimization over Networks

In this paper we develop random block coordinate descent methods for minimizing large-scale linearly constrained convex problems over networks. Since coupled constraints appear in the problem, we devise an algorithm that updates in parallel at each iteration at least two random components of the solution, chosen according to a given probability distribution. Those computations can be performed in a distributed fashion according to the structure of the network. Complexity per iteration of the proposed methods is usually cheaper than that of the full gradient method when the number of nodes in the network is much larger than the number of updated components. On smooth convex problems, we prove that these methods exhibit a sublinear worst-case convergence rate in the expected value of the objective function. Moreover, this convergence rate depends linearly on the number of components to be updated. On smooth strongly convex problems we prove that our methods converge linearly. We also focus on how to choose the probabilities to make our randomized algorithms converge as fast as possible, which leads us to solving a sparse semidefinite program. We then describe several applications that fit in our framework, in particular the convex feasibility problem. Finally, numerical experiments illustrate the behaviour of our methods, showing in particular that updating more than two components in parallel accelerates the method.     

Moduli for pointed convex domains

Summary A moduli space for the class of pointed strictly linearly convex domains in ⁿ is obtained. It is shown that the space of pointed smoothly bounded strictly linearly convex domains with a fixed indicatrix is parameterized by a class of deformations of the CR structure of the boundary of the indicatrix. These deformations are constructed by using the circular representation of a domain to pull back its complex structure tensor to the indicatrix. A careful study of the pull back structure shows that the allowable deformations are parameterized by a class of complex Hamiltonian vector fields. The proof of this fact is based on the Folland-Stein estimates for the complex of the boundary of the indicatrix.The paper is related to one of László Lempert, Holomorphic invariants, normal forms and moduli space of convex domains. Ann. Math128, 47–78 (1988), where other modular data for pointed convex domains were constructed. A method of recovering Lempert's modular data from the deformation moduli is given.Oblatum 26-IX-1989 & 22-III-1990Partially supported by an NSERC grant.The second author wishes to thank the University of Toronto and the Mathematical Sciences Research Institute at Berkeley, where portions of the paper were written.     

A Note on the Alternating Direction Method of Multipliers

We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled variables. The alternating direction method of multipliers has been well studied in the literature for the special case m=2, while it remains open whether its convergence can be extended to the general case m≥3. This note shows the global convergence of this extension when the involved functions are further assumed to be strongly convex.     

A note on augmented Lagrangian-based parallel splitting method

We consider the linearly constrained separable convex minimization problem, whose objective function consists of the sum of \(m\) individual convex functions in the absence of any coupling variables. While augmented Lagrangian-based decomposition methods have been well developed in the literature for solving such problems, a noteworthy requirement of these methods is that an additional correction step is a must to guarantee their convergence. This note shows that a straightforward Jacobian decomposition of the augmented Lagrangian method is globally convergent if the involved functions are further assumed to be strongly convex.     

THE EXSISTENCE OF $\mu -HOLOMORPHIC$ SEPARATING FUNCTION ON BOUNDED SMOOTH DOMAINS

The Complex analysis of strongly pseudoconvex domains in C^n is rather well known.In this paper it is proved that for a bounded smoothly domain $\omega$ there is a new complex structure on it under which $\omega$ will locally become a strongly convex even though the point on b&\omega& is not a pseudoconvex point from the view of the original complex structure.Particularly if $\omega$ is a weakly pseudoconvex domain,the $\mu$ cam be ,ade siffocoemt;u c;pse tp tje progoma; cp,[;ex strictire.Therefore a lot of properties of strongly pseudoconvex domains will become true on weakly pseudoconvex domains,or general domains.For example,it is proved that there is a $\mu-holomorphic$ separatin function which is holomorphic under the new complex structure.     

Invariant convex bodies for strongly elliptic systems

We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies are obtained for linear systems without zero order term on bounded domains and quasilinear systems of special form on bounded domains and on a class of unbounded domains. These conditions are formulated in algebraic form. They describe relation between the geometry of the invariant convex body and the coefficients of the system. Next, necessary conditions, which are also sufficient, for the invariance of some convex bodies are found for elliptic homogeneous systems with constant coefficients in a half-space. The necessary conditions are derived by using a criterion on the invariance of convex bodies for normalized matrix-valued integral transforms also obtained in the paper. In contrast with the previous studies of invariant sets for elliptic systems, no a priori restrictions on the coefficient matrices are imposed.     

Behavior of Solutions to the Dirichlet Problem for Elliptic Systems in Convex Domains

We consider the Dirichlet problem for strongly elliptic systems of order 2m in convex domains. Under a positivity assumption on the Poisson kernel it is proved that the weak solution has bounded derivatives up to order m provided the outward unit normal has no big jumps on the boundary. In the case of second order symmetric systems in plane convex domains the boundedness of the first derivatives is proved without the assumption on the normal.     

On U(n)-invariant strongly convex complex Finsler metrics

Science China Mathematics - In this paper, we obtain a necessary and sufficient condition for a U(n)-invariant complex Finsler metric F on domains in ?n to be strongly convex, which also...     

Volume Approximations of Strongly Pseudoconvex Domains

In convex geometry, the Blaschke surface area measure on the boundary of a convex domain can be interpreted in terms of the complexity of approximating polyhedra. This approach is formulated in the holomorphic setting to establish an alternate interpretation of Fefferman’s hypersurface measure on boundaries of strongly pseudoconvex domains in \(\mathbb {C}^2\). In particular, it is shown that Fefferman’s measure can be recovered from the Bergman kernel of the domain.     

Solving Rectilinear Planar Location Problems with Barriers by a Polynomial Partitioning

This paper considers planar location problems with rectilinear distance and barriers where the objective function is any convex, nondecreasing function of distance. Such problems have a non-convex feasible region and a nonconvex objective function. Based on an equivalent problem with modified barriers, derived in a companion paper [3], the non convex feasible set is partitioned into a network and rectangular cells. The rectangular cells are further partitioned into a polynomial number of convex subcells, called convex domains, on which the distance function, and hence the objective function, is convex. Then the problem is solved over the network and convex domains for an optimal solution. Bounds are given that reduce the number of convex domains to be examined. The number of convex domains is bounded above by a polynomial in the size of the problem.     

Maxwell and Lamé eigenvalues on polyhedra

In a convex polyhedron, a part of the Lamé eigenvalues with hard simple support boundary conditions does not depend on the Lamé coefficients and coincides with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter s linked to the Lamé coefficients and the associated eigenmodes are the gradients of the Laplace–Dirichlet eigenfunctions. In a non‐convex polyhedron, such a splitting of the spectrum disappears partly or completely, in relation with the non‐H² singularities of the Laplace–Dirichlet eigenfunctions. From the Maxwell equations point of view, this means that in a non‐convex polyhedron, the spectrum cannot be approximated by finite element methods using H¹ elements. Similar properties hold in polygons. We give numerical results for two L‐shaped domains. Copyright © 1999 John Wiley & Sons, Ltd.     

Foliations by stationary disks of almost complex domains

We study the problem of existence of stationary disks for domains in almost complex manifolds. As a consequence of our results, we prove that any almost complex domain which is a small deformation of a strictly linearly convex domain D⊂Cn with standard complex structure admits a singular foliation by stationary disks passing through any given internal point. Similar results are given for foliations by stationary disks through a given boundary point.     

Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization

It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.