We present several embedding results for 3-graded Lie algebras and KKT algebras that are generated by two homogeneous elements of degrees 1 and −1. We also propose the canonical kernel function for a “universal Bergman kernel” which extends the usual Bergman kernel on a bounded symmetric domain to a group-valued function or, in terms of formal series, to an element in the formal completion of t...

This paper is aimed toward the definition of a new exact augmented Lagrangian function for two-sided inequality constrained problems. The distinguishing feature of this augmented Lagrangian function is that it employs only one multiplier for each two-sided constraint. We prove that stationary points, local minimizers and global minimizers of the exact augmented Lagrangian function correspond ex...

to obtain an approximate second-order KKT solution of the `p-norm models in polynomial time with a fixed error tolerance, and then test our `p-norm models on CRSP(1992-2013) and also S&P 500 (2008-2012) data. The empirical results illustrate that our `p-norm regularized models can generate portfolios of any desired sparsity with portfolio variance, portfolio return and Sharpe Ratio comparable t...

We address the iterative solution of KKT systems arising in the solution of convex quadratic programming problems. Two strictly related and well established formulations for such systems are studied with particular emphasis on the effect of preconditioning strategies on their relation. Constraint and augmented preconditioners are considered, and the choice of the augmentation matrix is discusse...

Nonnegative Matrix Factorization (NMF) can be used to approximate a large nonnegative matrix as a product of two smaller nonnegative matrices. This paper shows in detail how an NMF algorithm based on Newton iteration can be derived utilizing the general Karush-KuhnTucker (KKT) conditions for first-order optimality. This algorithm is suited for parallel execution on shared-memory systems. It was...

We present an interior proximal point method with Bregman distance, whose Bregman function is separable and the zone is the interior of the positive orthant, for solving the quasiconvex optimization problem under nonnegative constraints. We establish that the sequence generated by our algorithm is well defined and we prove convergence to a solution point when the sequence of parameters goes to ...

In this paper, a stochastic solid transportation problem (SSTP) is constructed where the demand of item at destinations are randomly distributed. Such SSTP formulated with profit maximization form containing selling revenue, cost and holding/shortage item. The proposed framed as nonlinear which optimized through Karush–Kuhn–Tucker (KKT) conditions Lagrangian function. primary model bifurcated i...

Mathematical modeling of market design issues in liberalized electricity markets often leads to mixed-integer nonlinear multilevel optimization problems for which no general-purpose solvers exist and which are intractable in general. In this work, we consider the problem of splitting a market area into a given number of price zones such that the resulting market design yields welfareoptimal out...

This paper presents a perfect duality theory and a complete set of solutions to nonconvex quadratic programming problems subjected to inequality constraints. By use of the canonical dual transformation developed recently, a canonical dual problem is formulated, which is perfectly dual to the primal problem in the sense that they have the same set of KKT points. It is proved that the KKT points ...

In each iteration of the interior point method (IPM) at least one linear system has to be solved. The main computational effort of IPMs consists in the computation of these linear systems. Solving the corresponding linear systems with a direct method becomes very expensive for large scale problems. In this thesis, we have been concerned with using an iterative method for solving the reduced KKT...