On the computational complexity of piecewise-linear homotopy algorithms

This paper describes the main aspects of the ”piecewise-linear homotopy method” for fixed point approximation proposed by Eaves and Saigal [Eaves, C. B. and Saigal, R., Homotopies for computation of fixed points on unbounded regions, Mathematical Programming, 3 (1972), No. 1, 225–237]. The implementation of the method is developed using the modern programming language C# and then is used for solving some unconstrained optimization problems. The PL homotopy algorithm appears to be more reliable than the classical Newton method in the case of the problem of finding a local minima for Schwefel’s function and other optimization problems.

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Piecewise Linear Valued CSPs Solvable by Linear Programming Relaxation

ACM Transactions on Computational Logic ◽

10.1145/3488721 ◽

2022 ◽

Vol 23 (1) ◽

pp. 1-35

Author(s):

Manuel Bodirsky ◽

Marcello Mamino ◽

Caterina Viola

Keyword(s):

Computational Complexity ◽

Polynomial Time ◽

Piecewise Linear ◽

Systematic Investigation ◽

Constraint Satisfaction Problems ◽

Cost Functions ◽

Finite Domain ◽

Linear Programming Relaxation ◽

Infinite Domain ◽

The Cost

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of the complexity of infinite-domain VCSPs with piecewise linear homogeneous cost functions. Such VCSPs can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by reducing the problem to a finite-domain VCSP which can be solved using the basic linear program relaxation. It follows that VCSPs for submodular PLH cost functions can be solved in polynomial time; in fact, we show that submodular PLH functions form a maximally tractable class of PLH cost functions.

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Mid-Surfaces of Profile-based Freeforms for Mold Design

Journal of Manufacturing Science and Engineering ◽

10.1115/1.2831206 ◽

1999 ◽

Vol 121 (2) ◽

pp. 202-207 ◽

Cited By ~ 7

Author(s):

A. Fischer ◽

A. Smolin ◽

G. Elber

Keyword(s):

Computational Complexity ◽

Piecewise Linear ◽

Parametric Representation ◽

Iterative Refinement ◽

New Approach ◽

And Topology ◽

Matching Technique ◽

Optimization Function ◽

Refinement Process ◽

High Computational Complexity

Mid-surfaces of complex thin objects are commonly used in CAD applications for the analysis of casting and injection molding. However, geometrical representation in CAD typically takes the form of a solid representation rather than a mid-surface; therefore, a process for extracting the mid-surface is essential. Contemporary methods for extracting mid-surfaces are based on numerical computations using offsetting techniques or Voronoi diagram processes where the data is discrete and piecewise linear. These algorithms usually have high computational complexity, and their accuracy is not guaranteed. Furthermore, the geometry and topology of the object are not always preserved. To overcome these problems, this paper proposes a new approach for extracting a mid-surface from a freeform thin object. The proposed method reduces the mid-surface problem into a parametrization problem that is based on a matching technique in which a nonlinear optimization function is defined and solved according to mid-surface criteria. Then, the resulting mid-surface is dictated by a reparametrization process. The algorithm is implemented for freeform ruled, swept, and rotational surfaces, that are commonly used in engineering products. Reducing the problem to the profile curves of these surfaces alleviates the computational complexity of the 3D case and restricts it to a 2D case. Error is controlled globally through an iterative refinement process that utilizes continuous symbolic computations on the parametric representation. The feasibility of the proposed method is demonstrated through several examples.

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A comparative study of the PL homotopy and BFGS methods for some nonsmooth optimization problems

Creative Mathematics and Informatics ◽

10.37193/cmi.2019.02.01 ◽

2019 ◽

Vol 28 (2) ◽

pp. 97-104

Author(s):

ANDREI BOZANTAN ◽

VASILE BERINDE

Keyword(s):

Fixed Points ◽

Nonsmooth Optimization ◽

Optimization Problems ◽

Piecewise Linear ◽

Smooth Functions ◽

Inexact Line Search ◽

Homotopy Algorithm ◽

Unconstrained Optimization Problems ◽

Piecewise Linear Homotopy ◽

Computation Of Fixed Points

We consider some non-smooth functions and investigate the numerical behavior of the Piecewise Linear Hompotopy (PLH) method ([Bozântan, A., An implementation of the piecewise-linear homotopy algorithm for the computation of fixed points, Creat. Math. Inform., 19 (2010), No.~2, 140–148] and [Bozântan, A. and Berinde, V., Applications of the PL homotopy algorithm for the computation of fixed points to unconstrained optimization problems, Creat. Math. Inform., 22 (2013), No. 1, 41–46]). We compare the PLH method with the BFGS with inexact line search, a quasi-Newton method, having some results reported in [Lewis, A. S. and Overton, M. L., Nonsmooth optimization via BFGS, submitted to SIAM J. Optimiz, (2009)]. For most of the considered cases, the characteristics of the PLH method are quite similar to the BFGS method, that is, the PLH method converges to local minimum values and the convergence rate seems to be linear with respect to the number of function evaluations, but we also identify some issues with the PLH method.

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Fast exact conformalization of the lasso using piecewise linear homotopy

Biometrika ◽

10.1093/biomet/asz046 ◽

2019 ◽

Vol 106 (4) ◽

pp. 749-764 ◽

Cited By ~ 1

Author(s):

J Lei

Keyword(s):

Piecewise Linear ◽

Real Data ◽

Efficient Computation ◽

Computationally Efficient ◽

Conformal Prediction ◽

Trade Off ◽

Main Challenge ◽

Input Sample ◽

Piecewise Linear Homotopy ◽

Single Input

Summary Conformal prediction is a general method that converts almost any point predictor to a prediction set. The resulting set retains the good statistical properties of the original estimator under standard assumptions, and guarantees valid average coverage even when the model is mis-specified. A main challenge in applying conformal prediction in modern applications is efficient computation, as it generally requires an exhaustive search over the entire output space. In this paper we develop an exact and computationally efficient conformalization of the lasso and elastic net. The method makes use of a novel piecewise linear homotopy of the lasso solution under perturbation of a single input sample point. As a by-product, we provide a simpler and better-justified online lasso algorithm, which may be of independent interest. Our derivation also reveals an interesting accuracy-stability trade-off in conformal inference, which is analogous to the bias-variance trade-off in traditional parameter estimation. The practical performance of the new algorithm is demonstrated in both synthetic and real data examples.

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ANALYSIS OF THE COMPUTATIONAL COMPLEXITY OF THE METHOD OF ITERATIVE DIMENSIONAL-LINEAR GENERATION OF THE TRAJECTORY OF MOTION OF THE THREE-LINE ANTHROPOMORPHIC MANIPULATOR IN THE VOLUME SPACE WITH OBSTRUCTION

Proceedings of Southwest State University ◽

10.21869/2223-1560-2018-22-3-13-28 ◽

2018 ◽

Vol 22 (3) ◽

pp. 13-28 ◽

Cited By ~ 2

Author(s):

V. O. Antonov

Keyword(s):

Computational Complexity ◽

Real Time ◽

Piecewise Linear ◽

Recursive Algorithm ◽

Scheduling Algorithm ◽

Development Stage ◽

Target Point ◽

High Complexity ◽

Actual Problem ◽

Variable Parameters

Energy efficiency is an actual problem of the present, including in the field of robotics. Existing methods for planning the trajectory of motion of manipulators with excessive mobility face a number of problems, one of which is the impossibility of working in real time mode due to the high complexity of the scheduling algorithm. Moreover, the existing algorithms that work in real time are significantly inferior to the accuracy of the target operations. Therefore, earlier, in the author's articles, an iterative method of piecewise linear generation of the manipulator's trajectory was developed. In this paper, we analyze the computational complexity of the numerical method of iterative piecewise linear generation of the trajectory of a three-link anthropomorphic manipulator with 7 degrees of mobility in a volume space with an obstacle, an approximated hypersphere, in real time. A short description of the proposed method of planning the trajectory of motion is given. To move between the waypoints, the Denavite-Hartenberg representation used, with the formulation and solution of the problem of nonlinear optimization with the objective function of minimizing energy consumption when the manipulator moved to the target point. The initial generalized algorithm of the path planning method described. The number of operations that must performed in the process of execution of a recursive algorithm is considered. Parallelizing the branching recursive algorithm allows you to reduce the execution time to the time of executing a non-branching recursive algorithm with the same computational complexity and depth. A formula developed that allows you to select the values of variable parameters of the algorithm based on the available computational power and the allowable calculation time, and to determine the requirements for the manipulator computer system at the development stage.

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