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In this paper, we consider a feasible primal–dual interior point method for linear semidefinite programming problem ( S D P ) based on Alizadeh–Haeberly–Overton ( AHO ) direction (Monteiro, 1997). Firstly, and by a new and simple technique, we establish the existence and uniqueness of optimal solution of the perturbed problem ( S D P ) μ and its convergence to optimal solution of ( S D P ) . Next, we present new different alternatives to calculate the displacement step. After, we establish the convergence of the obtained algorithm and we show that its complexity is O ( n ln [ ε − 1 ( 〈 X 0 , S 0 〉 ) ] ) . Finally, we present some numerical simulations which show the effectiveness of the algorithm developed in this work. [ABSTRACT FROM AUTHOR]

Existing principal curve algorithms have some drawbacks such as time consuming and narrow application scope in practice, since these algorithms are mainly based on global optimization. In this paper, we present the concept of Constraint Local Principal Curve (CLPC), which uses local optimization methods and restricts the principal curve with two fixed endpoints to reduce the computational complexity. In addition, we propose three CLPC algorithms by Local Optimization and Adaptive Radius to expand the range of applications and increase the solution quality. The first algorithm, i.e., CLPCg is based on greedy thinking. The second algorithm, i.e., CLPCs uses one dimensional search and the last algorithm CLPCc combines the greedy thinking and one dimensional search. Then, we define six performance indices to evaluate the performance of the CLPC algorithms. Finally, we present some numerical experiments with three simulation data sets and two GPS measured data sets in both highway and railway. The results indicate that all of the three CLPC algorithms can obtain high-accuracy data from multiple low-accuracy data efficiently. The CLPC algorithms can improve the accuracy and computational speed compared with the existing K-segment principal curve (KPC) algorithm. In addition, CLPCc outperforms CLPCg and CLPCs according to the comprehensive experiments while CLPCg runs much faster than other ones. [ABSTRACT FROM AUTHOR]