Abstract:
In this paper, an alternating direction nonmonotone approximate Newton algorithm (ADNAN) based on nonmonotone line search is developed for solving inverse problems. It is shown that ADNAN converges to a solution of the inverse problems and numerical results provide the effectiveness of the proposed algorithm.

Abstract:
We examine possibility to design an efficient solving algorithm for problems of the class \np. It is introduced a classification of \np problems by the property that a partial solution of size $k$ can be extended into a partial solution of size $k+1$ in polynomial time. It is defined an unique class problems to be worth to search an efficient solving algorithm. The problems, which are outside of this class, are inherently exponential. We show that the Hamiltonian cycle problem is inherently exponential.

Abstract:
The minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming algorithms. When the number of variables is large, one of the most widely used strategies is to project the original problem into a small dimensional subspace. In this paper, we introduce an algorithm for solving nonlinear least squares problems. This algorithm is based on constructing a basis for the Krylov subspace in conjunction with a model trust region technique to choose the step. The computational step on the small dimensional subspace lies inside the trust region. The Krylov subspace is terminated such that the termination condition allows the gradient to be decreased on it. A convergence theory of this algorithm is presented. It is shown that this algorithm is globally convergent.

Abstract:
Quantum computation has attracted much attention since it was shown by Shor and Grover the possibility to implement quantum algorithms able to realize, respectively, factoring and searching in a faster way than any other known classical algorithm. It is possible to use Grover algorithm, taking profit of its ability to find a specific value in a unordered database, to find, for example, the zero of a logical function; the minimal or maximal value in a database or to recognize if an odd number is prime or not. Here we show quantum algorithms to solve those cited mathematical problems. The solution requires the use of a quantum bit string comparator being used as oracle. This quantum circuit compares two quantum states and identifies if they are equal or, otherwise, which of them is the largest. Moreover, we also show the quantum bit string comparator allow us to implement conditional statements in quantum computation, a fundamental structure for designing of algorithms.

Abstract:
We establish a quasi-Newton algorithm for solving a class of variational inequality problems which subproblems are linear equations. By presenting a suitable line search, the algorithm is well-defined. And under certain conditions, we get its global convergence and locally superlinear convergence.

Abstract:
This paper describes an original approach for determining independent loops needed for mesh-current analysis in order to solve circuit equation system arising in inductive Partial Element Equivalent Circuit (PEEC) approach. Based on the combined used of several simple algorithms, it considerably speed-up the loops search and enables the building of an associated matrix system with an improved condition number. The approach is so well-suited for large degrees of freedom problems, saving significantly memory and decreasing the time of resolution.

Abstract:
The broadcast scheduling problem (BSP) in packet radio networks is a well-known NP-complete combinatorial optimization problem. The broadcast scheduling avoids packet collisions by allowing only one node transmission in each collision domain of a time division multiple access (TDMA) network. It also improves the transmission utilization by assigning one transmission time slot to one or more nodes; thus, each node transmits at least once in each time frame. An optimum transmission schedule could minimize the length of a time frame while minimizing the number of idle nodes. In this paper, we propose a new iterated local search (ILS) algorithm that consists of two special perturbation and local search operators to solve the BSPs. Computational experiments are applied to benchmark data sets and randomly generated problem instances. The experimental results show that our ILS approach is effective in solving the problems with only a few runtimes, even for very large networks with 2,500 nodes.

Abstract:
The broadcast scheduling problem (BSP) in packet radio networks is a well-known NP-complete combinatorial optimization problem. The broadcast scheduling avoids packet collisions by allowing only one node transmission in each collision domain of a time division multiple access (TDMA) network. It also improves the transmission utilization by assigning one transmission time slot to one or more nodes; thus, each node transmits at least once in each time frame. An optimum transmission schedule could minimize the length of a time frame while minimizing the number of idle nodes. In this paper, we propose a new iterated local search (ILS) algorithm that consists of two special perturbation and local search operators to solve the BSPs. Computational experiments are applied to benchmark data sets and randomly generated problem instances. The experimental results show that our ILS approach is effective in solving the problems with only a few runtimes, even for very large networks with 2,500 nodes.

Abstract:
The Resource-Constrained Project Scheduling Problem (RCPSP) is a NP-hard problem in information engineering. The activities of a project have to be scheduled for satisfying all the precedence and resource constraints. We presented a heuristic algorithm (EDAS) to deal with this problem which employed an estimation of distribution algorithm (known as EDA) and improved the local search capacity with a simplex search. In this algorithm, the EDA firstly searched the solution space and generated activity lists to provide the initial population; then, the EDA selected the sample solutions to build a probability distribution model. The new individual was generated by sampling this model. The simplex search was used to enhance the local search capacity of the EDA. Compared with state-of-the-art algorithms available in the literature, we showed the effectiveness of this approach empirically on the standard benchmark problems of size J60 and J120 from PSPLIB.

Abstract:
Firstly, we give the Karush-Kuhn-Tucker (KKT) optimality condition of primal problem and introduce Jordan algebra simply. On the basis of Jordan algebra, we extend smoothing Fischer-Burmeister (F-B) function to Jordan algebra and make the complementarity condition smoothing. So the first-order optimization condition can be reformed to a nonlinear system. Secondly, we use the mixed line search quasi-Newton method to solve this nonlinear system. Finally, we prove the globally and locally superlinear convergence of the algorithm. 1. Introduction Linear second-order cone programming (SOCP) problems are convex optimization problems which minimize a linear function over the intersection of an affine linear manifold with the Cartesian product of second-order cones. Linear programming (LP), Linear second-order cone programming (SOCP), and semidefinite programming (SDP) all belong to symmetric cone analysis. LP is a special example of SOCP and SOCP is a special case of SDP. SOCP can be solved by the corresponding to the algorithm of SDP, and SOCP also has effectual solving method. Nesterov and Todd [1, 2] had an earlier research on primal-dual interior point method. In the rescent, it gives quick development about the solving method for SOCP. Many scholars concentrate on SOCP. The primal and dual standard forms of the linear SOCP are given by where the second-order cone : where refers to the standard Euclidean norm. In this paper, the vectors , , and and the matrix are partitioned conformally, namely Except for interior point method, semismoothing and smoothing Newton method can also be used to solve SOCP. In [3], the Karush-Kuhn-Tucker (KKT) optimality condition of primal-dual problem was reformulated to a semi-smoothing nonlinear system, which was solved by Newton method with central path. In [4], the KKT optimality condition of primal-dual problem was reformed to a smoothing nonlinear equations, then it was solved by combining Newton method with central path. References [3, 4] gave globally and locally quadratic convergent of the algorithm. 2. Preliminaries and Algorithm In this section, we introduce the Jordan algebra and give the nonlinear system, which comes from the Karush-Kuhn-Tucker (KKT) optimality condition. At last, we introduce two kinds of derivative-free line search rules. Associated with each vector , there is an arrow-shaped matrix which is defined as follows: Euclidean Jordan algebra is associated with second-order cones. For now we assume that all vectors consist of a single block . For two vectors and , define the following multiplication: