On the convergence of a class of outer approximation algorithms for convex programs

AbstractThis paper presents a new class of outer approximation methods for solving general convex programs. The methods solve at each iteration a subproblem whose constraints contain the feasible set of the original problem. Moreover, the methods employ quadratic objective functions in the subproblems by adding a simple quadratic term to the objective function of the original problem, while other outer approximation methods usually use the original objective function itself throughout the iterations. By this modification, convergence of the methods can be proved under mild conditions. Furthermore, it is shown that generalized versions of the cut construction schemes in Kelley-Cheney-Goldstein's cutting plane method and Veinott's supporting hyperplane method can be incorporated with the present methods and a cut generated at each iteration need not be retained in the succeeding iterations

Spectroscopic and Photochemical Research on the Diazo-Compounds etc.

(1) The photochemical absorption spectra of the diazo-compounds were measured, and the rays which act chemically were elucidated. (2) The photolysis of D. N. S. and D. B. S.-solutions was observed quantitatively by employing the light-thermostat. (Concentration of the solution was mol/200.) It was observed that the decomposition proceeds with the constant velocity independent of the concentration. It was shown theoretically that the influence of the decomposition product of D. N. S. upon the photolysis is negligible. (3) The influence of quinine sulphate upon the decomposition velocity of D. N. S. was observed. By adding quinine sulphate the decomposition velocity of D. N. S. was reduced. And the more quinine sulphate was added, the slower became the velocity. The result was compared with those from the kinetic equation which was derived by applying Langendyk-Weigert's distribution theory. The experimental values of _x/a were rather larger than those obtained from the equation ; and the result was discussed. (4) The influence of temperature upon the photolysis of D. N. S. and D. B. S. was studied, and the nature of the decomposition was made clear. (I) Temperature influence upon D. N. S. is negligible. The photolysis is practically a pure photochemical reaction. (II) Temperature influence upon D. B. S.-photolysis is also negligible in the region of 2°C and 22°C. The results obtained at 45°C and 60°C showed some complexity. In order to make clear the nature of this photolysis, the thermal decomposition of the solution was thoroughly studied. From the theoretical point of view, it was shown numerically how the pure photo-reaction of D. B. S. is disturbed by the thermal reaction, and how the decomposition is accelerated, together with its practical meaning. (5) It was found by measurement that the quantumefficiency of the photolysis of D. N. S.-solution at λ = 366 mμ was 0.16

Application of Fenchel's Duality Theorem to Penalty Methods in Convex Programming

This paper studies a new class of sequential unconstrained optimization methods, called the conjugate penalty method, for solving convex programming problems. The validity of the method is based on Fenchel's duality theorem. It is shown that, under certain condi- tions, conjugate penalty founctins are uniformly bounded on a neighborhood of a point which is an optimum of Fenchel's dual problem

A BLOCK-PARALLEL CONJUGATE GRADIENT METHOD FOR SEPARABLE QUADRATIC PROGRAMMING PROBLEMS1

Abstract For a large-scale quadratic programming problem with separable objective function, a variant of the conjugate gradient method can effectively be applied to the dual problem. In this paper, we consider a block-parallel modification of the conjugate gradient method, which is suitable for implementation on a parallel computer. More precisely, the method proceeds in a block Jacobi manner and executes the conjugate gradient iteration to solve quadratic programming subproblems associated with respective blocks. We implement the method on a Connection Machine Model CM-5 in the Single-Program Multiple-Data model of computation. We report some numerical results, which show that the proposed method is effective particularly for problems with some block structure