scholarly journals MINKOWSKI SUM SELECTION AND FINDING

2011 ◽  
Vol 21 (03) ◽  
pp. 283-311
Author(s):  
CHENG-WEI LUO ◽  
HSIAO-FEI LIU ◽  
PENG-AN CHEN ◽  
KUN-MAO CHAO
Keyword(s):  
Positive Integer ◽  
Objective Function ◽  
Real Number ◽  
Randomized Algorithms ◽  
Selection Problem ◽  
Minkowski Sum ◽  
Objective Functions ◽  
Deterministic Algorithms ◽  
Objective Value ◽  
Linear Objective Functions

Let P,Q ⊆ ℝ2 be two n-point multisets and Ar ≥ b be a set of λ inequalities on x and y, where A ∈ ℝλ×2, [Formula: see text], and b ∈ ℝλ. Define the constrained Minkowski sum(P ⊕ Q)Ar≥ b as the multiset {(p + q)|p ∈ P, q ∈ Q,A(p + q) ≥ b }. Given P, Q, Ar ≥ b , an objective function f : ℝ2 → ℝ, and a positive integer k, the MINKOWSKI SUM SELECTION problem is to find the kth largest objective value among all objective values of points in (P ⊕ Q)Ar≥ b . Given P, Q, Ar ≥ b , an objective function f : ℝ2 → ℝ, and a real number δ, the MINKOWSKI SUM FINDING problem is to find a point (x*, y*) in (P ⊕ Q)Ar≥ b such that |f(x*,y*) - δ| is minimized. For the MINKOWSKI SUM SELECTION problem with linear objective functions, we obtain the following results: (1) optimal O(n log n)-time algorithms for λ = 1; (2) O(n log 2 n)-time deterministic algorithms and expected O(n log n)-time randomized algorithms for any fixed λ > 1. For the MINKOWSKI SUM FINDING problem with linear objective functions or objective functions of the form [Formula: see text], we construct optimal O(n log n)-time algorithms for any fixed λ ≥ 1. As a byproduct, we obtain improved algorithms for the LENGTH-CONSTRAINED SUM SELECTION problem and the DENSITY FINDING problem.

Scalable Computing Architecture for Time-Dependent Transportation Optimization Problems Based on High-Performance Computing Techniques

2019 ◽  
Vol 2673 (4) ◽  
pp. 205-216 ◽  
Author(s):  
Pengfei (Taylor) Li ◽  
Peirong (Slade) Wang ◽  
Farzana Chowdhury ◽  
Li Zhang
Keyword(s):  
Objective Function ◽  
High Performance Computing ◽  
High Performance ◽  
Optimization Problems ◽  
Dynamic Network ◽  
Alternative Formulation ◽  
High Fidelity ◽  
Objective Functions ◽  
Performance Computing ◽  
Transportation Optimization

Traditional formulations for transportation optimization problems mostly build complicating attributes into constraints while keeping the succinctness of objective functions. A popular solution is the Lagrangian decomposition by relaxing complicating constraints and then solving iteratively. Although this approach is effective for many problems, it generates intractability in other problems. To address this issue, this paper presents an alternative formulation for transportation optimization problems in which the complicating attributes of target problems are partially or entirely built into the objective function instead of into the constraints. Many mathematical complicating constraints in transportation problems can be efficiently modeled in dynamic network loading (DNL) models based on the demand–supply equilibrium, such as the various road or vehicle capacity constraints or “IF–THEN” type constraints. After “pre-building” complicating constraints into the objective functions, the objective function can be approximated well with customized high-fidelity DNL models. Three types of computing benefits can be achieved in the alternative formulation: ( a) the original problem will be kept the same; ( b) computing complexity of the new formulation may be significantly reduced because of the disappearance of hard constraints; ( c) efficiency loss on the objective function side can be mitigated via multiple high-performance computing techniques. Under this new framework, high-fidelity and problem-specific DNL models will be critical to maintain the attributes of original problems. Therefore, the authors’ recent efforts in enhancing the DNL’s fidelity and computing efficiency are also described in the second part of this paper. Finally, a demonstration case study is conducted to validate the new approach.

scholarly journals Intramax and other objective functions: The case of Slovenia

2016 ◽  
Vol 24 (2) ◽  
pp. 12-25 ◽  
Author(s):  
Samo Drobne ◽  
Mitja Lakner
Keyword(s):  
Objective Function ◽  
The Other ◽  
Objective Functions ◽  
Interaction Index ◽  
Functional Regions ◽  
Hierarchical Aggregation ◽  
Administrative Units

Abstract The use of different objective functions in hierarchical aggregation procedures is examined in this paper. Specifically, we analyse the use of the original Intramax objective function, the sum-of-flows objective function, the sum-of-proportions-to-intra-regional-flows objective function, Smart’s weighted interaction index, the first and second CURDS weighted interaction indices, and Tolbert and Killian’s interaction index. The results of the functional regionalisation have been evaluated by self-containment statistics, and they show that the use of the original Intramax procedure tends to delineate operationally the most persuasive and balanced regions that, regarding the intra-regional flows, homogeneously cover the analysed territory. The other objective functions give statistically better but operationally less suitable results. Functional regions modelled using the original Intramax procedure were compared to the regions at NUTS 2 and NUTS 3 levels, as well as to administrative units in Slovenia. We conclude that there are some promising directions for further research on functional regionalisation using hierarchical aggregation procedures.

Development of a Stepped Calibration Approach for XAJ Hydrological Model

2013 ◽  
Vol 405-408 ◽  
pp. 2222-2225
Author(s):  
Qian Li ◽  
Wei Min Bao ◽  
Jing Lin Qian
Keyword(s):  
Root Mean Square Error ◽  
Objective Function ◽  
Traditional Method ◽  
Hydrological Model ◽  
Model Performance ◽  
Layer By Layer ◽  
Mean Square ◽  
Objective Functions ◽  
Single Objective ◽  
Calibration Approach

This paper discusses the conceptual stepped calibration approach (SCA) which has been developed for the Xinanjiang (XAJ) model. Multi-layer and multi-objective functions which can make optimization work simpler and more effective are introduced in this procedure. In all eight parameters were considered, they were divided into four layers according to the structure of XAJ model, and then calibrated layer by layer. The SCA procedure tends to improve the performance of the traditional method of calibration (thus, using a single objective function, such as root mean square error RMSE). The compared results demonstrate that the SCA yield better model performance than RMSE.

scholarly journals Lane-deviation penalty formulation and analysis for autonomous vehicle avoidance maneuvers

2021 ◽  
pp. 095440702110079
Author(s):  
Pavel Anistratov ◽  
Björn Olofsson ◽  
Lars Nielsen
Keyword(s):  
Objective Function ◽  
Traffic Safety ◽  
Autonomous Vehicles ◽  
Autonomous Vehicle ◽  
Emergency Situation ◽  
Change Scenario ◽  
Initial Speed ◽  
Objective Functions ◽  
Double Lane Change ◽  
Tire Friction

Autonomous vehicles hold promise for increased vehicle and traffic safety, and there are several developments in the field where one example is an avoidance maneuver. There it is dangerous for the vehicle to be in the opposing lane, but it is safe to drive in the original lane again after the obstacle. To capture this basic observation, a lane-deviation penalty (LDP) objective function is devised. Based on this objective function, a formulation is developed utilizing optimal all-wheel braking and steering at the limit of road–tire friction. This method is evaluated for a double lane-change scenario by computing the resulting behavior for several interesting cases, where parameters of the emergency situation such as the initial speed of the vehicle and the size and placement of the obstacle are varied, and it performs well. A comparison with maneuvers obtained by minimum-time and other lateral-penalty objective functions shows that the use of the considered penalty function decreases the time that the vehicle spends in the opposing lane.

A new representation for a robot grasping quality measure

Robotica ◽  
1995 ◽  
Vol 13 (3) ◽  
pp. 287-295 ◽  
Author(s):  
Venugopal K. Varma ◽  
Uri Tasch
Keyword(s):  
Objective Function ◽  
Material Properties ◽  
Graphical Representation ◽  
Contact Forces ◽  
Geometric Constraints ◽  
Objective Functions ◽  
Finger Force ◽  
Finger Forces ◽  
Robot Grasping ◽  
Selection Of

SummaryWhen an object is held by a multi-fingered hand, the values of the contact forces can be multivalued. An objective function, when used in conjunction with the frictional and geometric constraints of the grasp, can however, give a unique set of finger force values. The selection of the objective function in determining the finger forces is dependent on the type of grasp required, the material properties of the object, and the limitations of the röbot fingers. In this paper several optimization functions are studied and their merits highlighted. The paper introduces a graphical representation of the finger force values and the objective functions that enable one to select and compare various grasping configurations. The impending motion of the object at different torque and finger force values are determined by observing the normalized coefficient of friction plots.

On the kth root partition function

2021 ◽  
pp. 1-15
Author(s):  
Ya-Li Li ◽  
Jie Wu
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Real Number ◽  
Number Of Solutions ◽  
Asymptotic Development ◽  
Number Formula

For any positive integer [Formula: see text], let [Formula: see text] be the number of solutions of the equation [Formula: see text] with integers [Formula: see text], where [Formula: see text] is the integral part of real number [Formula: see text]. Recently, Luca and Ralaivaosaona gave an asymptotic formula for [Formula: see text]. In this paper, we give an asymptotic development of [Formula: see text] for all [Formula: see text]. Moreover, we prove that the number of such partitions is even (respectively, odd) infinitely often.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

Cooperative Coevolution with Formula-Based Variable Grouping for Large-Scale Global Optimization

2018 ◽  
Vol 26 (4) ◽  
pp. 569-596 ◽  
Author(s):  
Yuping Wang ◽  
Haiyan Liu ◽  
Fei Wei ◽  
Tingting Zong ◽  
Xiaodong Li
Keyword(s):  
Global Optimization ◽  
Objective Function ◽  
Real World ◽  
Large Scale ◽  
Divide And Conquer ◽  
Group Method ◽  
Objective Functions ◽  
Elementary Functions ◽  
Cooperative Coevolution ◽  
Solution Quality

For a large-scale global optimization (LSGO) problem, divide-and-conquer is usually considered an effective strategy to decompose the problem into smaller subproblems, each of which can then be solved individually. Among these decomposition methods, variable grouping is shown to be promising in recent years. Existing variable grouping methods usually assume the problem to be black-box (i.e., assuming that an analytical model of the objective function is unknown), and they attempt to learn appropriate variable grouping that would allow for a better decomposition of the problem. In such cases, these variable grouping methods do not make a direct use of the formula of the objective function. However, it can be argued that many real-world problems are white-box problems, that is, the formulas of objective functions are often known a priori. These formulas of the objective functions provide rich information which can then be used to design an effective variable group method. In this article, a formula-based grouping strategy (FBG) for white-box problems is first proposed. It groups variables directly via the formula of an objective function which usually consists of a finite number of operations (i.e., four arithmetic operations “[Formula: see text]”, “[Formula: see text]”, “[Formula: see text]”, “[Formula: see text]” and composite operations of basic elementary functions). In FBG, the operations are classified into two classes: one resulting in nonseparable variables, and the other resulting in separable variables. In FBG, variables can be automatically grouped into a suitable number of non-interacting subcomponents, with variables in each subcomponent being interdependent. FBG can easily be applied to any white-box problem and can be integrated into a cooperative coevolution framework. Based on FBG, a novel cooperative coevolution algorithm with formula-based variable grouping (so-called CCF) is proposed in this article for decomposing a large-scale white-box problem into several smaller subproblems and optimizing them respectively. To further enhance the efficiency of CCF, a new local search scheme is designed to improve the solution quality. To verify the efficiency of CCF, experiments are conducted on the standard LSGO benchmark suites of CEC'2008, CEC'2010, CEC'2013, and a real-world problem. Our results suggest that the performance of CCF is very competitive when compared with those of the state-of-the-art LSGO algorithms.

scholarly journals An evolutionary-based optimization algorithm for truss sizing design

2016 ◽  
Vol 38 (4) ◽  
pp. 307-317
Author(s):  
Pham Hoang Anh
Keyword(s):  
Local Search ◽  
Objective Function ◽  
Optimization Algorithm ◽  
Optimization Problems ◽  
Optimal Solution ◽  
The Other ◽  
Truss Structures ◽  
Benchmark Problems ◽  
Discrete Variables ◽  
Objective Functions

In this paper, the optimal sizing of truss structures is solved using a novel evolutionary-based optimization algorithm. The efficiency of the proposed method lies in the combination of global search and local search, in which the global move is applied for a set of random solutions whereas the local move is performed on the other solutions in the search population. Three truss sizing benchmark problems with discrete variables are used to examine the performance of the proposed algorithm. Objective functions of the optimization problems are minimum weights of the whole truss structures and constraints are stress in members and displacement at nodes. Here, the constraints and objective function are treated separately so that both function and constraint evaluations can be saved. The results show that the new algorithm can find optimal solution effectively and it is competitive with some recent metaheuristic algorithms in terms of number of structural analyses required.

scholarly journals Performance of Static Iot Networks using RPL Objective Functions

2019 ◽  
Vol 8 (4) ◽  
pp. 8972-8977 ◽  
Keyword(s):  
Objective Function ◽  
Routing Protocol ◽  
Routing Algorithms ◽  
Transmission Protocol ◽  
Objective Functions ◽  
Lossy Networks ◽  
Hop Count ◽  
Efficient Data ◽  
Selection Of ◽  
Better Than

Internet of Things, abbreviated as IoT is a network used mainly for the communication where different devices are connected for the retrieval, examination and execution of the necessary task. One of IoT’s biggest challenge is that, they are resource-constrained. Hence, it is essential to use an efficient data transmission protocol for routing. An effective routing protocol for static IoT network is the Routing protocol for Low Power and Lossy Networks (RPL). It is essential to assess the effectiveness of the RPL with the selection of best objective function for different static model. In this paper, the performance of different routing algorithms is compared in connection with different static topologies. Hence, the objective function’s performance is compared for different topologies i.e., Butterfly, Ring and Umbrella topologies. We consider two objective functions: namely Minimum Rank with Hysteresis Objective Function (MRHOF) and Objective Function Zero (OF0). MRHOF considers Expected Transmission Count (ETX) as its metric and the metric considered under OF0 is hop count. It is observed that the objective function OF0 performs better than MRHOF for the metric of energy and successful receiving of data.

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