Simplex-based Algorithms Research Articles

Simplex Guided Extremum Seeking Control With Convergence Detection to Improve Global Performance

This paper presents a novel approach to improve the global performance of extremum seeking control (ESC). This algorithm employs perturbation-based ESC to find local extrema and a simplex-based algorithm to search for the global extremum. The two methods are employed in an iterative switching manner. We propose a novel frequency-domain criterion to detect when the state of a system under ESC has converged to a local optimum. Switching from ESC to the simplex method is triggered when this criterion is satisfied. We prove that this criterion will be met within finite time, under conditions on the ESC law. Simulations and experiments demonstrate and statistically prove that this algorithm outperforms ESC or simplex-based algorithms for optimizing objective functions with multiple extrema.

基于单形体几何的高光谱遥感图像解混算法

A new simplex geometry-based algorithm is proposed to estimate abundance images for hyperspectral unmixing. With a priori knowledge of endmember signatures, the algorithm is designed to find the abundance value corresponding to each endmember at each observation pixel. Under the linear spectral mixture model, hyperspectral unmixing can be considered as a convex geometry problem, in which the endmembers are located in the vertices of simplex enclosing the hyperspectral data set and the barycentric coordinates of observation pixels with respect to the simplex corresponding to the abundances of endmembers. The proposed algorithm consists of three parts: simplex volume-based methods to calculate the barycentric coordinates, an algorithm which solves the distance geometry constraint problem, and subspace determination by an algorithm based on the barycenter of simplex. Compared with the other simplex-based algorithms, the proposed method has several advantages. The Cayley-Menger matrix is introduced to convert the computation among pixels into the computation involved in the pairwise distances between them, which give a more accurate result with a low computational complexity as well as a good conservation about the geometrical construction. Meanwhile, the use of barycenter of simplex builds an accurate and efficient method to judge the subspaces containing the estimated point. Then a recursive method is developed to get the estimated abundances. In addition, only the distances between the observation pixels and the endmembers are involved in the algorithm and so a dimensionality reduction transform is not necessary in this algorithm, which can save from the loss of useful information during the dimensionality reduction. Experimental results on synthetic and real hyperspectral datasets demonstrate that the proposed algorithm has a more accurate result compared with the state-of-the-art algorithms, fully constrained least squares (FCLS) and simplex-projection unmixing (SPU), and it is less time-consuming when the number of endmembers is small.

A new approach based on orthogonal bases of data space to decomposition of mixed pixels for hyperspectral imagery

A new algorithm for decomposition of mixed pixels based on orthogonal bases of data space is proposed in this paper. It is a simplex-based method which extracts endmembers sequentially using computations of largest simplex volumes. At each searching step of this extraction algorithm, searching for the simplex with the largest volume is equivalent to searching for a new orthogonal basis which has the largest norm. The new endmember corresponds to the new basis with the largest norm. This algorithm runs very fast and can also avoid the dilemma in traditional simplex-based endmember extraction algorithms, such as N-FINDR, that it generally produces different sets of final endmembers if different initial conditions are used. Moreover, with this set of orthogonal bases, the proposed algorithm can also determine the proper number of endmembers and finish the unmixing of the original images which the traditional simplex-based algorithms cannot do by themselves. Experimental results of both artificial simulated images and practical remote sensing images demonstrate the algorithm proposed in this paper is a fast and accurate algorithm for the decomposition of mixed pixels.

Towards a closer integration of finite domainpropagation and simplex-based algorithms

This paper describes our experience in implementing an industrial application using thefinite domain solver of the ECL i PS e constraint logic programming (CLP) system, inconjunction with the mathematical programming (MP) system, FortMP. In this technique,the ECL i PS e system generates a feasible solution that is adapted to construct a starting point(basic solution) for the MP solver. The basic solution is then used as an input to the FortMPsystem to warm-start the simplex (SX) algorithm, hastening the solution of the linearprogramming relaxation, (LPR). SX proceeds as normal to find the optimal integer solution.Preliminary results indicate that the integration of the two environments is suitable for thisapplication in particular, and may generally yield significant benefits. We describe adaptationsrequired in the hybrid method, and report encouraging experimental results for thisproblem.

K-Sum Linear Programming

In this note we introduce the k-sum linear programming problem (KLP) which subsumes the classical linear programming problem and the minmax linear programming problem. KLP can be transformed into a linear program with an exponential number of additional constraints and one additional variable. Exploiting the special structure of these additional constraints, we show that KLP can be solved in polynomial time. Two promising simplex-based algorithms are also suggested to solve KLP.