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Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5. Keywords: non-linear and linear entanglement witnesses PACS number(s): 03.67.Mn, 03.65.Ud

Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5. Keywords: non-linear and linear entanglement witnesses PACS number(s): 03.67.Mn, 03.65.Ud

Quasi-one-dimensional quantum structures with spectra scaling faster than the square of the eigenmode number (superscaling) can generate intrinsic, off-resonant optical nonlinearities near the fundamental physical limits, independent of the details of the potential energy along the structure. The scaling of spectra is determined by the topology of the structure, while the magnitudes of the transition moments are set by the geometry of the structure. This paper presents a comprehensive study of the geometrical optimization of superscaling quasi-one-dimensional structures and provides heuristics for designing molecules to maximize intrinsic response. A main result is that designers of conjugated structures should attach short side groups at least a third of the way along the bridge, not near its end as is conventionally done. A second result is that once a side group is properly placed, additional side groups do not further enhance the response.

New higher order methods for linear programming have been developed that have led to significant improvement in performance.

We examine a multidimensional optimization problem in the tropical mathematics setting. The problem is to minimize a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield under linear inequality constraints. Given two arbitrary matrices, the objective function is defined by one of the matrices through the use of multiplicative conjugate transposition, whereas the inequality constraints are given by the other matrix. We start with an overview of known tropical optimization problems with both linear and nonlinear objective functions, subject to linear equality and inequality constraints. A short introduction to tropical algebra is given to provide a formal framework to solve the optimization problem under study. As a preliminary result, a general solution to a linear inequality with an arbitrary matrix is presented. We describe an example optimization problem drawn from project scheduling and then offer a general representation of the problem. Furthermore, we introduce an additional unknown variable and reduce the problem to solving a linear inequality, where the new variable plays the role of a parameter. A necessary and sufficient condition for the inequality to hold is used to evaluate the parameter, whereas the solution of the inequality is taken as a solution for the optimization problem. As a result, under fairly general conditions, a complete direct solution to the problem is derived in a compact vector form. Numerical and graphical examples for two-dimensional problems are given to illustrate obtained solutions.

A globally convergent primal-dual predictor corrector interior point algorithm for linear programming has been developed.

Three major objectives were completed during the year. The first demonstrates how to directly use rank-one updates to a Cholesky factorization of the required inverse for Karmarkar projections while fully exploiting sparsity. This can significantly improve computational speed when only a few variables are changing significantly at each step. The second demonstrates a new method for adding new variables to a quasi-Newton Hessian approximation which preserves problem scale and positive definiteness of the Hessian. Numerical results show the method to be preferable to known methods. The third examines a variety of ways of implementing a sequential quadratic programming code, and uses numerical testing to indicate a suitable merit function and good algorithms for updating Lagrange multiplier and Hessian approximations. Recent new results for updating Hessians for unconstrained problems are currently being studied to determine if better Hessian approximations can be obtained.

The project has been concerned with developing new numerical techniques to solve large scale linear and nonlinear programming problems. Early work focused on sequential quadratic programming techniques for nonlinear programming. Subsequently, all work was focused on interior point methods for large scale linear and nonlinear programming. Initially, the focus of the research was on both dual-affine and primal-dual algorithm for linear programming. Substantial computational experience demonstrated the superiority of the primal-dual methods, and subsequent research focused on improving the efficiency of these methods, both by adding higher order methods via predictor- corrector techniques and by improving the linear algebra to take advantage of both sparsity and machine architecture. Most recently, research has focused on large scale quadratic programming. A primal-dual predictor-corrector method has been devised and shown to be very promising computationally for problems with diagonal or sparsely structured Hessian matrices. For problems with dense Hessians, a pure primal conjugate projected gradient algorithm shows promise on small problems. It remains to be tested on large-scale problems.