Approximate solutions for two-level optimization problems

1988 ◽  
pp. 181-196 ◽  
Author(s):  
P. Loridan ◽  
J. Morgan
Keyword(s):  
Optimization Problems ◽  
Approximate Solutions

scholarly journals A Dai-Liao Hybrid Hestenes-Stiefel and Fletcher-Revees Methods for Unconstrained Optimization

2021 ◽  
Vol 2 (1) ◽  
pp. 33
Author(s):  
Nasiru Salihu ◽  
Mathew Remilekun Odekunle ◽  
Also Mohammed Saleh ◽  
Suraj Salihu
Keyword(s):  
Unconstrained Optimization ◽  
Line Search ◽  
Optimization Problems ◽  
Convex Combination ◽  
Gradient Methods ◽  
Approximate Solutions ◽  
Step Size ◽  
Conjugacy Condition ◽  
Inexact Line Search ◽  
Wolfe Line Search

Some problems have no analytical solution or too difficult to solve by scientists, engineers, and mathematicians, so the development of numerical methods to obtain approximate solutions became necessary. Gradient methods are more efficient when the function to be minimized continuously in its first derivative. Therefore, this article presents a new hybrid Conjugate Gradient (CG) method to solve unconstrained optimization problems. The method requires the first-order derivatives but overcomes the steepest descent method’s shortcoming of slow convergence and needs not to save or compute the second-order derivatives needed by the Newton method. The CG update parameter is suggested from the Dai-Liao conjugacy condition as a convex combination of Hestenes-Stiefel and Fletcher-Revees algorithms by employing an optimal modulating choice parameterto avoid matrix storage. Numerical computation adopts an inexact line search to obtain the step-size that generates a decent property, showing that the algorithm is robust and efficient. The scheme converges globally under Wolfe line search, and it’s like is suitable in compressive sensing problems and M-tensor systems.

On quasi approximate solutions for nonsmooth robust semi-infinite optimization problems

2019 ◽  
Vol 35 (3) ◽  
pp. 417-426 ◽  
Author(s):  
CHANOKSUDA KHANTREE ◽  
RABIAN WANGKEEREE ◽  
◽  
Keyword(s):  
Constraint Qualification ◽  
Generalized Convexity ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Data Uncertainty ◽  
Necessary Condition ◽  
Duality Theorems ◽  
Approximate Form ◽  
Locally Lipschitz ◽  
Approximate Duality

This paper devotes to the quasi ε-solution for robust semi-infinite optimization problems (RSIP) involving a locally Lipschitz objective function and infinitely many locally Lipschitz constraint functions with data uncertainty. Under the fulfillment of robust type Guignard constraint qualification and robust type Kuhn-Tucker constraint qualification, a necessary condition for a quasi ε-solution to problem (RSIP). After introducing the generalized convexity, we give a sufficient optimality for such a quasi ε-solution to problem (RSIP). Finally, we also establish approximate duality theorems in term of Wolfe type which is formulated in approximate form.

scholarly journals A non-exponential extension of Sanov’s theorem via convex duality

2020 ◽  
Vol 52 (1) ◽  
pp. 61-101
Author(s):  
Daniel Lacker
Keyword(s):  
Optimal Transport ◽  
Large Deviation ◽  
Optimization Problems ◽  
Risk Measures ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Dual Pair ◽  
Wasserstein Distance ◽  
Empirical Measures ◽  
Empirical Distributions

AbstractThis work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

scholarly journals Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems

2019 ◽  
Vol 5 (4) ◽  
pp. eaav2372 ◽  
Author(s):  
Hayato Goto ◽  
Kosuke Tatsumura ◽  
Alexander R. Dixon
Keyword(s):  
Combinatorial Optimization ◽  
Hamiltonian Systems ◽  
Large Scale ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Combinatorial Optimization Problems ◽  
Bifurcation Phenomena ◽  
Field Programmable ◽  
Max Cut Problem ◽  
Adiabatic Optimization

Combinatorial optimization problems are ubiquitous but difficult to solve. Hardware devices for these problems have recently been developed by various approaches, including quantum computers. Inspired by recently proposed quantum adiabatic optimization using a nonlinear oscillator network, we propose a new optimization algorithm simulating adiabatic evolutions of classical nonlinear Hamiltonian systems exhibiting bifurcation phenomena, which we call simulated bifurcation (SB). SB is based on adiabatic and chaotic (ergodic) evolutions of nonlinear Hamiltonian systems. SB is also suitable for parallel computing because of its simultaneous updating. Implementing SB with a field-programmable gate array, we demonstrate that the SB machine can obtain good approximate solutions of an all-to-all connected 2000-node MAX-CUT problem in 0.5 ms, which is about 10 times faster than a state-of-the-art laser-based machine called a coherent Ising machine. SB will accelerate large-scale combinatorial optimization harnessing digital computer technologies and also offer a new application of computational and mathematical physics.

scholarly journals Evolutionary Algorithms in Intelligent Systems

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1733
Author(s):  
Alfredo Milani
Keyword(s):  
Evolutionary Algorithms ◽  
Intelligent Systems ◽  
Optimization Problems ◽  
Approximate Solutions

Evolutionary algorithms and metaheuristics are widely used to provide efficient and effective approximate solutions to computationally difficult optimization problems [...]

scholarly journals On approximate solutions in set-valued optimization problems

2012 ◽  
Vol 236 (17) ◽  
pp. 4421-4427 ◽  
Author(s):  
María Alonso-Durán ◽  
Luis Rodríguez-Marín
Keyword(s):  
Optimization Problems ◽  
Approximate Solutions

Computational Complexity Analysis of Selective Breeding Algorithm

2014 ◽  
Vol 591 ◽  
pp. 172-175
Author(s):  
M. Chandrasekaran ◽  
P. Sriramya ◽  
B. Parvathavarthini ◽  
M. Saravanamanikandan
Keyword(s):  
Computational Complexity ◽  
Selective Breeding ◽  
Heuristic Algorithms ◽  
Complexity Analysis ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Approximate Solutions ◽  
Problem Size ◽  
Combinatorial Optimization Problems ◽  
Complete Problems

In modern years, there has been growing importance in the design, analysis and to resolve extremely complex problems. Because of the complexity of problem variants and the difficult nature of the problems they deal with, it is arguably impracticable in the majority time to build appropriate guarantees about the number of fitness evaluations needed for an algorithm to and an optimal solution. In such situations, heuristic algorithms can solve approximate solutions; however suitable time and space complication take part an important role. In present, all recognized algorithms for NP-complete problems are requiring time that's exponential within the problem size. The acknowledged NP-hardness results imply that for several combinatorial optimization problems there are no efficient algorithms that realize a best resolution, or maybe a close to best resolution, on each instance. The study Computational Complexity Analysis of Selective Breeding algorithm involves both an algorithmic issue and a theoretical challenge and the excellence of a heuristic.

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