AN OPTIMAL ALGORITHM FOR COMPUTING (≤K)-LEVELS, WITH APPLICATIONS

1996 ◽  
Vol 06 (03) ◽  
pp. 247-261 ◽  
Author(s):  
HAZEL EVERETT ◽  
JEAN-MARC ROBERT ◽  
MARC VAN KREVELD
Keyword(s):  
Optimization Problems ◽  
Optimal Algorithm ◽  
Optimal Solution ◽  
Time Algorithm ◽  
Geometric Optimization ◽  
Line Transversal ◽  
Constant Number ◽  
Separation Problem ◽  
Line Segments ◽  
Weak Separation

This paper gives an optimal O(n log n+nk) time algorithm for constructing the levels 1,…, k in an arrangement of n lines in the plane. This algorithm is extended to compute these levels in an arrangement of n unbounded x-monotone polygonal convex chains, of which each pair intersects at most a constant number of times. We then show how these results can be used to solve several geometric optimization problems including the weak separation problem for sets of red and blue points or polygons, the maximum line transversal problem for sets of line segments, the densest hemisphere problem for sets of points on a sphere and the optimal corridor problem for sets of points in the plane. All of the algorithms are quality-sensitive; they run faster if the optimal solution is a good one.

scholarly journals Minimum Cell Connection in Line Segment Arrangements

2017 ◽  
Vol 27 (03) ◽  
pp. 159-176
Author(s):  
Helmut Alt ◽  
Sergio Cabello ◽  
Panos Giannopoulos ◽  
Christian Knauer
Keyword(s):  
Linear Time ◽  
Optimal Solution ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Constant Number ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Number Of Segments ◽  
Connection Problems

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segments are on the boundary of [Formula: see text]. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.

SIMPLE OPTIMAL ALGORITHMS FOR RECTILINEAR LINK PATH AND POLYGON SEPARATION PROBLEMS

1999 ◽  
Vol 09 (01) ◽  
pp. 31-42 ◽  
Author(s):  
ANIL MAHESHWARI ◽  
JÖRG-RÜDIGER SACK
Keyword(s):  
Parallel Algorithms ◽  
Random Access ◽  
Optimal Solution ◽  
Simple Polygon ◽  
Separation Problem ◽  
Line Segments ◽  
Point Pair ◽  
Erew Pram ◽  
Rectilinear Polygons ◽  
Minimum Number

The link metric, defined on a constrained region R of the plane, sets the distance between a pair of points in R to equal the minimum number of line segments or links needed to construct a path in R between the point pair. The minimum rectilinear link path problem considered here is to compute a rectilinear path consisting of the minimum number of links between two points in R, when R is inside an n-sided rectilinear simple polygon. In this paper we present optimal sequential and parallel algorithms to compute a minimum rectilinear link path in a trapezoided region R. Our parallel algorithm requires O( log n) time using a total of O(n) operations. The complexity of our algorithm matches that of the algorithm of McDonald and Peters [19]. By exploiting the dual structure of the trapezoidation of R, we obtain a conceptually simple and easy to implement algorithm. As applications of our techniques we provide an optimal solution to the minimum nested polygon problem and the minimum polygon separation problem. The minimum nested polygon problem asks for finding a rectilinear polygon, with minimum number of sides, that is nested between two given rectilinear polygons one of which is contained in the other. The minimum polygon separation problem asks for computing a minimum number of orthogonal lines and line segments that separate two given non-intersecting simple rectilinear polygons. All parallel algorithms are deterministic, designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM), and are optimal.

scholarly journals A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hamid Reza Erfanian ◽  
M. H. Noori Skandari ◽  
A. V. Kamyad
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Nonsmooth Equations ◽  
Smooth Functions ◽  
Generalized Derivative ◽  
Novel Approach

We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented.

Study on Mix-Variable Collaborative Design Optimization

2012 ◽  
Vol 215-216 ◽  
pp. 592-596
Author(s):  
Li Gao ◽  
Rong Rong Wang
Keyword(s):  
Design Optimization ◽  
Product Design ◽  
Optimization Algorithm ◽  
Collaborative Design ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Collaborative Optimization ◽  
Continuous Variables ◽  
Complex Product ◽  
Divide And Rule

In order to deal with complex product design optimization problems with both discrete and continuous variables, mix-variable collaborative design optimization algorithm is put forward based on collaborative optimization, which is an efficient way to solve mix-variable design optimization problems. On the rule of “divide and rule”, the algorithm decouples the problem into some relatively simple subsystems. Then by using collaborative mechanism, the optimal solution is obtained. Finally, the result of a case shows the feasibility and effectiveness of the new algorithm.

Application of High-Performance Computing to Numerical Simulation of Human Movement

1995 ◽  
Vol 117 (1) ◽  
pp. 155-157 ◽  
Author(s):  
F. C. Anderson ◽  
J. M. Ziegler ◽  
M. G. Pandy ◽  
R. T. Whalen
Keyword(s):  
Computer Architecture ◽  
High Performance ◽  
Large Scale ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Human Movement ◽  
Parallel Machine ◽  
Optimal Controls ◽  
Processing Machine ◽  
Vector Processing

We have examined the feasibility of using massively-parallel and vector-processing supercomputers to solve large-scale optimization problems for human movement. Specifically, we compared the computational expense of determining the optimal controls for the single support phase of gait using a conventional serial machine (SGI Iris 4D25), a MIMD parallel machine (Intel iPSC/860), and a parallel-vector-processing machine (Cray Y-MP 8/864). With the human body modeled as a 14 degree-of-freedom linkage actuated by 46 musculotendinous units, computation of the optimal controls for gait could take up to 3 months of CPU time on the Iris. Both the Cray and the Intel are able to reduce this time to practical levels. The optimal solution for gait can be found with about 77 hours of CPU on the Cray and with about 88 hours of CPU on the Intel. Although the overall speeds of the Cray and the Intel were found to be similar, the unique capabilities of each machine are better suited to different portions of the computational algorithm used. The Intel was best suited to computing the derivatives of the performance criterion and the constraints whereas the Cray was best suited to parameter optimization of the controls. These results suggest that the ideal computer architecture for solving very large-scale optimal control problems is a hybrid system in which a vector-processing machine is integrated into the communication network of a MIMD parallel machine.

An Electrochemical Model Based Optimal Charging Algorithm for Lithium-Ion Batteries

2015 ◽  
Author(s):  
Sourav Pramanik ◽  
Sohel Anwar
Keyword(s):  
Management System ◽  
Optimization Problems ◽  
Optimal Algorithm ◽  
Algorithm Design ◽  
Lithium Ion ◽  
Cell Voltage ◽  
Realistic Model ◽  
Constant Current ◽  
Battery Management System ◽  
Model Based

In recent years, Lithium-Ion battery has gathered lot of importance in many forms of energy storage applications due to its overwhelming benefits. Any battery pack alone cannot achieve its optimal performance unless there is a robust and efficient energy management system, commonly known as battery management system or BMS. The Lithium-Ion charger is a voltage-limiting device that is similar to the lead acid system. The difference lies in a higher cell voltage; tighter voltage tolerance and the absence of trickle or float charge at full charge. In this work, we propose the design of a novel optimal strategy for charging the battery that better suits the battery performance. A performance index is defined that aims at minimizing the effort of regeneration along with a minimum deviation from the rated maximum thresholds for cell temperature and charging current. A more realistic model based on battery electrochemistry is used for the optimal algorithm design as opposed to equivalent circuit models. To solve the optimization problem, Pontryagin’s principle is used which is very effective for constrained optimization problems with both state and input constraints. Simulation results show that the proposed optimal charging algorithm is capable of shortening the charging time of a Lithium Ion cell while maintaining the temperature constraint when compared with the standard constant current charging.

A Novel Approach to Designing Surrogate-assisted Genetic Algorithms by Combining Efficient Learning of Walsh Coefficients and Dependencies

2021 ◽  
Vol 1 (2) ◽  
pp. 1-23
Author(s):  
Arkadiy Dushatskiy ◽  
Tanja Alderliesten ◽  
Peter A. N. Bosman
Keyword(s):  
Boolean Functions ◽  
Surrogate Model ◽  
Optimization Problems ◽  
State Of The Art ◽  
Fitness Function ◽  
Optimal Solution ◽  
Problem Structure ◽  
Novel Approach ◽  
Efficient Learning ◽  
Nk Landscapes

Surrogate-assisted evolutionary algorithms have the potential to be of high value for real-world optimization problems when fitness evaluations are expensive, limiting the number of evaluations that can be performed. In this article, we consider the domain of pseudo-Boolean functions in a black-box setting. Moreover, instead of using a surrogate model as an approximation of a fitness function, we propose to precisely learn the coefficients of the Walsh decomposition of a fitness function and use the Walsh decomposition as a surrogate. If the coefficients are learned correctly, then the Walsh decomposition values perfectly match with the fitness function, and, thus, the optimal solution to the problem can be found by optimizing the surrogate without any additional evaluations of the original fitness function. It is known that the Walsh coefficients can be efficiently learned for pseudo-Boolean functions with k -bounded epistasis and known problem structure. We propose to learn dependencies between variables first and, therefore, substantially reduce the number of Walsh coefficients to be calculated. After the accurate Walsh decomposition is obtained, the surrogate model is optimized using GOMEA, which is considered to be a state-of-the-art binary optimization algorithm. We compare the proposed approach with standard GOMEA and two other Walsh decomposition-based algorithms. The benchmark functions in the experiments are well-known trap functions, NK-landscapes, MaxCut, and MAX3SAT problems. The experimental results demonstrate that the proposed approach is scalable at the supposed complexity of O (ℓ log ℓ) function evaluations when the number of subfunctions is O (ℓ) and all subfunctions are k -bounded, outperforming all considered algorithms.

Adaptive Grouping Brain Storm Optimization for Multimodal Optimization Problems

2021 ◽  
Vol 12 (4) ◽  
pp. 81-100
Author(s):  
Yao Peng ◽  
Zepeng Shen ◽  
Shiqi Wang
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Optimal Algorithm ◽  
Peak Detection ◽  
Multimodal Optimization ◽  
Optimal Solutions ◽  
Grouping Strategy ◽  
Different Dimensions ◽  
Global Optimal ◽  
Localization Ability

Multimodal optimization problem exists in multiple global and many local optimal solutions. The difficulty of solving these problems is finding as many local optimal peaks as possible on the premise of ensuring global optimal precision. This article presents adaptive grouping brainstorm optimization (AGBSO) for solving these problems. In this article, adaptive grouping strategy is proposed for achieving adaptive grouping without providing any prior knowledge by users. For enhancing the diversity and accuracy of the optimal algorithm, elite reservation strategy is proposed to put central particles into an elite pool, and peak detection strategy is proposed to delete particles far from optimal peaks in the elite pool. Finally, this article uses testing functions with different dimensions to compare the convergence, accuracy, and diversity of AGBSO with BSO. Experiments verify that AGBSO has great localization ability for local optimal solutions while ensuring the accuracy of the global optimal solutions.

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