scholarly journals On the Tightness of Semidefinite Relaxations for Rotation Estimation

2021 ◽  
Author(s):  
Lucas Brynte ◽  
Viktor Larsson ◽  
José Pedro Iglesias ◽  
Carl Olsson ◽  
Fredrik Kahl
Keyword(s):  
General Framework ◽  
Optimization Problems ◽  
Extreme Points ◽  
Objective Functions ◽  
Semidefinite Relaxations ◽  
Theoretical Understanding ◽  
Convex Optimization Problems ◽  
Empirical Performance ◽  
Estimation Problems ◽  
And Robotics

AbstractWhy is it that semidefinite relaxations have been so successful in numerous applications in computer vision and robotics for solving non-convex optimization problems involving rotations? In studying the empirical performance, we note that there are few failure cases reported in the literature, in particular for estimation problems with a single rotation, motivating us to gain further theoretical understanding. A general framework based on tools from algebraic geometry is introduced for analyzing the power of semidefinite relaxations of problems with quadratic objective functions and rotational constraints. Applications include registration, hand–eye calibration, and rotation averaging. We characterize the extreme points and show that there exist failure cases for which the relaxation is not tight, even in the case of a single rotation. We also show that some problem classes are always tight given an appropriate parametrization. Our theoretical findings are accompanied with numerical simulations, providing further evidence and understanding of the results.

scholarly journals Multiplicative Approximations, Optimal Hypervolume Distributions, and the Choice of the Reference Point

2015 ◽  
Vol 23 (1) ◽  
pp. 131-159 ◽  
Author(s):  
Tobias Friedrich ◽  
Frank Neumann ◽  
Christian Thyssen
Keyword(s):  
Evolutionary Algorithms ◽  
Reference Point ◽  
Pareto Front ◽  
Optimization Problems ◽  
Extreme Points ◽  
Approximation Ratio ◽  
Objective Functions ◽  
Theoretical Understanding ◽  
Multi Objective ◽  
Trade Offs

Many optimization problems arising in applications have to consider several objective functions at the same time. Evolutionary algorithms seem to be a very natural choice for dealing with multi-objective problems as the population of such an algorithm can be used to represent the trade-offs with respect to the given objective functions. In this paper, we contribute to the theoretical understanding of evolutionary algorithms for multi-objective problems. We consider indicator-based algorithms whose goal is to maximize the hypervolume for a given problem by distributing [Formula: see text] points on the Pareto front. To gain new theoretical insights into the behavior of hypervolume-based algorithms, we compare their optimization goal to the goal of achieving an optimal multiplicative approximation ratio. Our studies are carried out for different Pareto front shapes of bi-objective problems. For the class of linear fronts and a class of convex fronts, we prove that maximizing the hypervolume gives the best possible approximation ratio when assuming that the extreme points have to be included in both distributions of the points on the Pareto front. Furthermore, we investigate the choice of the reference point on the approximation behavior of hypervolume-based approaches and examine Pareto fronts of different shapes by numerical calculations.

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 Decentralized and parallel primal and dual accelerated methods for stochastic convex programming problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Darina Dvinskikh ◽  
Alexander Gasnikov
Keyword(s):  
Convex Optimization ◽  
Inverse Problems ◽  
Convex Programming ◽  
Data Science ◽  
Optimization Problems ◽  
Stochastic Gradient ◽  
Logarithmic Factor ◽  
Accelerated Methods ◽  
Convex Optimization Problems ◽  
Primal And Dual

Abstract We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles, the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique, we show that the proposed methods with stochastic oracle can be additionally parallelized at each node. The considered algorithms can be applied to many data science problems and inverse problems.

scholarly journals NeuSE: A Neural Snapshot Ensemble Method for Collaborative Filtering

2021 ◽  
Vol 15 (6) ◽  
pp. 1-20
Author(s):  
Dongsheng Li ◽  
Haodong Liu ◽  
Chao Chen ◽  
Yingying Zhao ◽  
Stephen M. Chu ◽  
...  
Keyword(s):  
Collaborative Filtering ◽  
Optimization Problems ◽  
Empirical Studies ◽  
Large Datasets ◽  
Model Learning ◽  
Global Models ◽  
Convex Optimization Problems ◽  
Memory Network ◽  
Real World Datasets ◽  
Performance Tradeoff

In collaborative filtering (CF) algorithms, the optimal models are usually learned by globally minimizing the empirical risks averaged over all the observed data. However, the global models are often obtained via a performance tradeoff among users/items, i.e., not all users/items are perfectly fitted by the global models due to the hard non-convex optimization problems in CF algorithms. Ensemble learning can address this issue by learning multiple diverse models but usually suffer from efficiency issue on large datasets or complex algorithms. In this article, we keep the intermediate models obtained during global model learning as the snapshot models, and then adaptively combine the snapshot models for individual user-item pairs using a memory network-based method. Empirical studies on three real-world datasets show that the proposed method can extensively and significantly improve the accuracy (up to 15.9% relatively) when applied to a variety of existing collaborative filtering methods.

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