Convexity of Nonlinear Image of a Small Ball with Applications to Optimization

Let f: XY be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f (x) is Lipschitz in B(a,r) with Lipschitz constant L and f (a) is a surjection: f (a)X=Y; this implies the existence of >0 such that f (a)^* yy, yY. Then, if r,/(2L), the image F=f(B(a,)) of the ball B(a,) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint x–a. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for small power control is convex. This leads to various results in optimal control.     

Log-Sigmoid Multipliers Method in Constrained Optimization

In this paper we introduced and analyzed the Log-Sigmoid (LS) multipliers method for constrained optimization. The LS method is to the recently developed smoothing technique as augmented Lagrangian to the penalty method or modified barrier to classical barrier methods. At the same time the LS method has some specific properties, which make it substantially different from other nonquadratic augmented Lagrangian techniques.We established convergence of the LS type penalty method under very mild assumptions on the input data and estimated the rate of convergence of the LS multipliers method under the standard second order optimality condition for both exact and nonexact minimization.Some important properties of the dual function and the dual problem, which are based on the LS Lagrangian, were discovered and the primal–dual LS method was introduced.     

Newton’s method and its use in optimization

Newton’s method is a basic tool in numerical analysis and numerous applications, including operations research and data mining. We survey the history of the method, its main ideas, convergence results, modifications, its global behavior. We focus on applications of the method for various classes of optimization problems, such as unconstrained minimization, equality constrained problems, convex programming and interior point methods. Some extensions (non-smooth problems, continuous analog, Smale’s results, etc.) are discussed briefly, while some others (e.g., versions of the method to achieve global convergence) are addressed in more details.     

Nonlinear rescaling and proximal-like methods in convex optimization

The nonlinear rescaling principle (NRP) consists of transforming the objective function and/or the constraints of a given constrained optimization problem into another problem which is equivalent to the original one in the sense that their optimal set of solutions coincides. A nonlinear transformation parameterized by a positive scalar parameter and based on a smooth sealing function is used to transform the constraints. The methods based on NRP consist of sequential unconstrained minimization of the classical Lagrangian for the equivalent problem, followed by an explicit formula updating the Lagrange multipliers. We first show that the NRP leads naturally to proximal methods with an entropy-like kernel, which is defined by the conjugate of the scaling function, and establish that the two methods are dually equivalent for convex constrained minimization problems. We then study the convergence properties of the nonlinear rescaling algorithm and the corresponding entropy-like proximal methods for convex constrained optimization problems. Special cases of the nonlinear rescaling algorithm are presented. In particular a new class of exponential penalty-modified barrier functions methods is introduced. Partially supported by the National Science Foundation, under Grants DMS-9201297, and DMS-9401871. Partially supported by NASA Grant NAG3-1397 and NSF Grant DMS-9403218.     

A geometrical presentation of the surface mapping class group and surgery

We construct a tangle presentation of the mapping class group similar to a natural presentation of the braid group by geometrical braids. A relation between surgery and Heegard diagrams for 3-manifolds arising in this way and different applications are studied.Supported by Rotshild Fellowship     

Electronically excited states of membrane fluorescent probe 4-dimethylaminochalcone. Results of quantum chemical calculations

Quantum-chemical calculations of ground and excited states for membrane fluorescent probe 4-dimethylaminochalcone (DMAC) in vacuum were performed. Optimized geometries and dipole moments for lowest-lying singlet and triplet states were obtained. The nature of these electronic transitions and the relaxation path in the excited states were determined; changes in geometry and charge distribution were assessed. It was shown that in vacuum the lowest existed level is of (n, π*) nature, and the closest to it is the level of (π, π*) nature; the energy gap between them is narrow. This led to an effective (1)(π, π*) →(1)(n, π*) relaxation. After photoexcitation the molecule undergoes significant transformations, including changes in bond orders, pyramidalization angle of the dimethylamino group, and planarity of the molecule. Its dipole moment rises from 5.5 Debye in the ground state to 17.1 Debye in the (1)(π, π*) state, and then falls to 2 Debye in the (1)(n, π*) state. The excited (1)(n, π*) state is a short living state; it has a high probability of intersystem crossing into the (3)(π, π*) triplet state. This relaxation path explains the low quantum yield of DMAC fluorescence in non-polar media. It is possible that (3)(π, π*) is responsible for observed DMAC phosphorescence.     

Cubic regularization of Newton method and its global performance

In this paper, we provide theoretical analysis for a cubic regularization of Newton method as applied to unconstrained minimization problem. For this scheme, we prove general local convergence results. However, the main contribution of the paper is related to global worst-case complexity bounds for different problem classes including some nonconvex cases. It is shown that the search direction can be computed by standard linear algebra technique.     

Absolute configuration of diastereomeric derivatives of N-substituted aziridine-2-carboxylic acids

Individual diastereomers of a number of anilides of N-substituted aziridine-2-carboxylic acids have been obtained. Conclusions regarding the spatial structure of the compounds indicated have been drawn on the basis of the ¹H NMR, ¹⁵N NMR, and CD spectra.Report 37 of the series entitled Asymmetric nonbridging nitrogen. For report 36 see [1].Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 10, pp. 1343–1348, October, 1984.     

Synthesis and lactonization of 1-(2-hydroxy-1,1-dimethylethyl)-aziridine-2-carboxylic acid esters

The reaction of 2,3-dibromopropanoates with 2-amino-2-methyl-1-propanol in the presence of triethylamine gave a number of 1-(2-hydroxy-1,1-dimethylethyl)aziridine-2-carboxylic acid esters, which, under the influence of basic catalysts, were converted to a bicyclic lactone — 2,2-dimethyl-4-oxa-1-azabicyclo[4.1.0]heptan-5-one. The effect of the structure of the substrate and the nature of the lactonizing agent on the rate of cyclization was studied. Two new catalysts for the cyclization of hydroxy esters to lactones, viz., CsF/Al₂O₃ and Cs₂CO₃/18-crown-6, are proposed.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 10, pp. 1340–1343, October, 1988.