In this paper, We define JORDAN ALGEBRA bundles of finite type and we give one-one correspondence between JORDAN ALGEBRA bundles of finite type and JORDAN rings.

A full Nesterov-Todd (NT) step INFEASIBLE INTERIOR-POINT algorithm is proposed for solving monotone linear complementarity problems over SYMMETRIC cones by using EUCLIDEAN JORDAN ALGEBRA. Two types of full NT-steps are used, feasibility steps and centering steps. The algorithm starts from strictly feasible iterates of a perturbed problem, and, using the central path and feasibility steps, finds strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, strictly feasible iterates are obtained to be close enough to the central path of the new perturbed problem. The starting point depends on two positive numbers rp and rd. The algorithm terminates either by finding an e-solution or detecting that the SYMMETRIC cone linear complementarity problem has no optimal solution with vanishing duality gap satisfying a condition in terms of rp and rd. The iteration bound coincides with the best known bound for INFEASIBLE INTERIOR-POINT methods.

In this paper, by using the exponential convexity property of a barrier function, we propose an INFEASIBLE INTERIOR-POINT method for Cartesian P_*(k) horizontal linear complementarity problem over SYMMETRIC cones. The method uses Nesterov and Todd full steps, and we prove that the proposed algorithm is well define. The iteration bound coincides with the currently best iteration bound for the Cartesian P_*(k) horizontal linear complementarity problem over SYMMETRIC cones.

The search for finding the local minimization in unconstrained optimization problems and a fixed point of the gradient system of ordinary differential equations are two close problems. Limited-memory algorithms are widely used to solve large-scale problems, while Rang Kuta's methods are also used to solve numerical differential equations. In this paper, using the concept of sub-space method and fixed-step length and integration of line-search and trust-region techniques, an ODE-based hybrid method is proposed for solving large-scale optimization problems. Since the line-search methods may require more iteration for convergence, while Trust-region methods also require a lot of iteration to solve the constrained sub problem, a new class of methods is proposed in this way, which combines the best features of trust-region and line-search methods. The main feature of the proposed method is that the linear equation system is solved only once in order to obtain the experimental step.

In this paper, we propose a feasible INTERIOR-POINT algorithm for mixed SYMMETRIC cone linear complementarity problems which are a general class of complementarity problems. The symmetrization of the search directions used in this paper is based on Nesterov and Todd scaling scheme. By using EUCLIDEAN JORDAN ALGEBRA, we prove the convergence analysis of the proposed algorithm and show that the complexity bound of the algorithm matches the currently best known iteration bound for feasible INTERIOR-POINT methods.

In this paper, we present a second-order corrector INFEASIBLE INTERIOR-POINT method for linear optimization in a large neighborhood of the central path. The innovation of our method is to calculate the predictor directions using a speci c kernel function instead of the logarithmic barrier function. We decompose the predictor direction induced by the kernel function to two orthogonal directions of the corresponding to the negative and positive component of the right-hand side vector of the centering equation. The method then considers the new point as a linear combination of these directions along with a second-order corrector direction. The convergence analysis of the proposed method is investigated and it is proved that the complexity bound is O(n 5 4 log " 1 ).

Objective: Risk management is one of the most important areas of study in finance, and its vital role in the field has attracted the attention of managers and investors in in various sectors of the industry. Especially in recent years, with the onset of financial crises, the importance and necessity of accurate studies in this area has doubled. The main purpose of this study is to provide a model for a more accurate measurement of equity portfolio risk. Methods: To conduct this research, adjusted closing prices of a sample of thirty listed companies have been used. CVaR is the main model and four other models are formulated using different methods of variance modeling. The first method calculates conditional value at risk using constant variance and in the other three methods, variance is modeled on GARCH, E-GARCH and T-GARCH approaches. Results: Ultimately, the results have been evaluated using appropriate statistical tests, namely paired t test and Wilcoxon signed rank test. The results obtained from both tests suggest that the method used to model variance has a significant effect on attaining a better optimal portfolio. Conclusion: Considering results of the research, which approve the tested hypothesis, one can conclude, taking into account the heteroscedasticity in Iranian capital market, would result in a better optimized portfolio. Moreover, the results illustrated that the use of CVaR model, for risk measurement, rather than previously used traditional models, can be effective in improving the performance, in optimizing stock portfolios, significantly.