Application of the network equivalent concept for external system representation for power system transient analysis is well known. However, the challenge to utilize an equivalent network, approximated by a rational function, is to guarantee the passivity of the corresponding model. In this regard, special techniques are required to enforce the passivity of the equivalent model through a post processing approach that minimizes its impact on the original model characteristics. In this paper, the passivity is enforced by expressing the problem in terms of a Convex Optimization problem that guarantees the global optimal solution. The Convex Optimization problem is efficiently solved by recently developed numerical interior–point methods. This passivity enforcement is also global which indicates that the passivity enforcement in one region does not lead to passivity violation in other regions.

Design of sparse array antenna that can create the desired radiation patterns with minimum number of elements, is a favorite research area. The synthesis sparse array problem can be modeled with appropriate constraints on the number of solve space members, namely l_0-norm of the weight elements. But it is a non-Convex problem that requires to solving a NP-hard problem. An interesting ideas is mentioned to relax problem to Convex problem. The proposed solution is based l_1-norm,The algorithm used here, first determines the optimal radiation pattern with Convex Optimization. then by using iterative weighting l_1-norm, sparse array is obtained by removing those elements that weights of them are almost zero and optimally determines the position of the element. As a result, by solving the non-Convexity property of the problem, the optimal solution is provided with a reasonable computational time. The purpose of the Optimization method is to minimize the number of elements, observe the constraints related to the requirements of the radiation pattern and reduce the calculation time. This research, in its case study, was able to sparse the 11×11 array (121 elements) to 42 elements (increase PSL) and 37 elements (increase mainlobe beamwidth) by adjusting the relevant parameters such as DRR, γ and ε.

The problem of finding the minimum cost multi-commodity flow in an undirected and complete network is studied when the link costs are piecewise linear and Convex. The arc-path model and overflow model are presented to formulate the problem. The results suggest that the new overflow model outperforms the classical arc-path model for this problem. The classical revised simplex, Frank and Wolf and a heuristic method are compared for the problem.

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‎The Proximal Stochastic Average Gradient (Prox-SAG+) is a primary method used for solving Optimization problems that contain the sum of two Convex functions. This kind of problem usually arises in machine learning, which utilizes a large amount of data to create component functions from a dataset. A proximal operation is applied to obtain the optimal value due to its appropriate properties. The Prox-SAG+ algorithm is faster than some other methods and has a simpler algorithm than previous ones. Moreover, using this specific operator can help to reassure that the achieved result is optimal. Additionally, it has been proven that the proposed method has an approximately geometric rate of convergence. Implementing the proposed operator makes the method more practical than other algorithms found in the literature. Numerical analysis also confirms the efficiency of the proposed scheme.

In this paper, an algorithm is proposed to improve the constrained PID control design based on the Convex-concave Optimization. The control system is designed by optimizing a performance cost function, taking into account the stability and efficiency constraints with frequency domain analysis in which the sensitivity and complementary sensitivity concepts are used. It is shown, using a counter example, the previous methods are not effective for some systems, the Optimization problem becomes unbounded and interrupted. To solve the problem, conditions where the Optimization problem fails to have a response are analyzed and the previous limitations are eliminated by representing a new designing method. The performance of the proposed scheme is shown by applying it to the counter example. Moreover, the control system is designed for an unstable system.