scholarly journals Inexact accelerated high-order proximal-point methods

2021 ◽  
Author(s):  
Yurii Nesterov
Keyword(s):  
Rate Of Convergence ◽  
Optimization Problems ◽  
Auxiliary Problem ◽  
Unconstrained Minimization ◽  
High Order ◽  
Proximal Point ◽  
Lipschitz Continuous ◽  
Proximal Points ◽  
New Framework ◽  
Better Than

AbstractIn this paper, we present a new framework of bi-level unconstrained minimization for development of accelerated methods in Convex Programming. These methods use approximations of the high-order proximal points, which are solutions of some auxiliary parametric optimization problems. For computing these points, we can use different methods, and, in particular, the lower-order schemes. This opens a possibility for the latter methods to overpass traditional limits of the Complexity Theory. As an example, we obtain a new second-order method with the convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This rate is better than the maximal possible rate of convergence for this type of methods, as applied to functions with Lipschitz continuous Hessian. We also present new methods with the exact auxiliary search procedure, which have the rate of convergence $$O\left( k^{-(3p+1)/ 2}\right) $$ O k - ( 3 p + 1 ) / 2 , where $$p \ge 1$$ p ≥ 1 is the order of the proximal operator. The auxiliary problem at each iteration of these schemes is convex.

scholarly journals Local convergence of tensor methods

2021 ◽  
Author(s):  
Nikita Doikov ◽  
Yurii Nesterov
Keyword(s):  
Local Convergence ◽  
Optimization Problems ◽  
High Order ◽  
Order Tensor ◽  
High Order Derivative ◽  
Complexity Bounds ◽  
Lipschitz Continuous ◽  
Tensor Methods ◽  
Smooth Component ◽  
Inexact Proximal

AbstractIn this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity bounds for the Composite Tensor Method in convex and uniformly convex cases are also discussed. Lastly, we show how local convergence of the methods can be globalized using the inexact proximal iterations.

scholarly journals Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1113
Author(s):  
Isaías Alonso-Mallo ◽  
Ana M. Portillo
Keyword(s):  
Numerical Experiments ◽  
Splitting Method ◽  
Method Of Lines ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
High Order ◽  
Time Dependent ◽  
Boundary Values ◽  
Lipschitz Continuous

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

scholarly journals The Rate of Convergence of Lupasq-Analogue of the Bernstein Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Heping Wang ◽  
Yanbo Zhang
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

We discuss the rate of convergence of the Lupasq-analogues of the Bernstein operatorsRn,q(f;x)which were given by Lupas in 1987. We obtain the estimates for the rate of convergence ofRn,q(f)by the modulus of continuity off, and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

Polynomial Moving Fitting Method for Edge Identification

2011 ◽  
Vol 308-310 ◽  
pp. 2560-2564 ◽  
Author(s):  
Xiang Rong Yuan
Keyword(s):  
Edge Detection ◽  
Curve Fitting ◽  
Polynomial Function ◽  
High Order ◽  
Polynomial Fitting ◽  
Fitting Method ◽  
Order Polynomial ◽  
Better Than ◽  
Low Order

A moving fitting method for edge detection is proposed in this work. Polynomial function is used for the curve fitting of the column of pixels near the edge. Proposed method is compared with polynomial fitting method without sub-segment. The comparison shows that even with low order polynomial, the effects of moving fitting are significantly better than that with high order polynomial fitting without sub-segment.

Improved Trust Region Based MPS Method for High-Dimensional Expensive Black-Box Problems

2013 ◽  
Author(s):  
George H. Cheng ◽  
Adel Younis ◽  
Kambiz Haji Hajikolaei ◽  
G. Gary Wang
Keyword(s):  
Optimization Problems ◽  
Trust Region ◽  
Black Box ◽  
High Dimensional ◽  
Test Problems ◽  
Design Variables ◽  
Low Dimensionality ◽  
Improve Algorithm ◽  
Improved Performance ◽  
Better Than

Mode Pursuing Sampling (MPS) was developed as a global optimization algorithm for optimization problems involving expensive black box functions. MPS has been found to be effective and efficient for problems of low dimensionality, i.e., the number of design variables is less than ten. A previous conference publication integrated the concept of trust regions into the MPS framework to create a new algorithm, TRMPS, which dramatically improved performance and efficiency for high dimensional problems. However, although TRMPS performed better than MPS, it was unproven against other established algorithms such as GA. This paper introduces an improved algorithm, TRMPS2, which incorporates guided sampling and low function value criterion to further improve algorithm performance for high dimensional problems. TRMPS2 is benchmarked against MPS and GA using a suite of test problems. The results show that TRMPS2 performs better than MPS and GA on average for high dimensional, expensive, and black box (HEB) problems.

scholarly journals Metaheuristic Optimization of Power and Energy Systems: Underlying Principles and Main Issues of the ‘Rush to Heuristics’

Energies ◽  
2020 ◽  
Vol 13 (19) ◽  
pp. 5097
Author(s):  
Gianfranco Chicco ◽  
Andrea Mazza
Keyword(s):  
Global Optimization ◽  
Specific Problem ◽  
Optimization Problems ◽  
Energy Systems ◽  
Progressive Increase ◽  
Metaheuristic Algorithms ◽  
Metaheuristic Algorithm ◽  
Power And Energy Systems ◽  
Power And Energy ◽  
Better Than

In the power and energy systems area, a progressive increase of literature contributions that contain applications of metaheuristic algorithms is occurring. In many cases, these applications are merely aimed at proposing the testing of an existing metaheuristic algorithm on a specific problem, claiming that the proposed method is better than other methods that are based on weak comparisons. This ‘rush to heuristics’ does not happen in the evolutionary computation domain, where the rules for setting up rigorous comparisons are stricter but are typical of the domains of application of the metaheuristics. This paper considers the applications to power and energy systems and aims at providing a comprehensive view of the main issues that concern the use of metaheuristics for global optimization problems. A set of underlying principles that characterize the metaheuristic algorithms is presented. The customization of metaheuristic algorithms to fit the constraints of specific problems is discussed. Some weaknesses and pitfalls that are found in literature contributions are identified, and specific guidelines are provided regarding how to prepare sound contributions on the application of metaheuristic algorithms to specific problems.

scholarly journals An algorithm for LC1 optimization

2005 ◽  
Vol 15 (2) ◽  
pp. 301-306 ◽  
Author(s):  
Nada Djuranovic-Milicic
Keyword(s):  
Rate Of Convergence ◽  
Unconstrained Optimization ◽  
Optimization Problems ◽  
Directional Derivative ◽  
Second Order ◽  
General Algorithm ◽  
Unconstrained Optimization Problems ◽  
Convergence Proof

In this paper an algorithm for LC1 unconstrained optimization problems, which uses the second order Dini upper directional derivative is considered. The purpose of the paper is to establish general algorithm hypotheses under which convergence occurs to optimal points. A convergence proof is given, as well as an estimate of the rate of convergence.

scholarly journals The Median Solution of the Newsvendor Problem and Some Observations

2015 ◽  
Vol 6 (3) ◽  
pp. 55-60
Author(s):  
Pritibhushan Sinha
Keyword(s):  
Standard Deviation ◽  
Theoretical Analysis ◽  
Numerical Experiment ◽  
Objective Function ◽  
Optimization Problems ◽  
Newsvendor Problem ◽  
Expected Profit ◽  
Random Optimization ◽  
Better Than

Abstract We consider the median solution of the Newsvendor Problem. Some properties of such a solution are shown through a theoretical analysis and a numerical experiment. Sometimes, though not often, median solution may be better than solutions maximizing expected profit, or maximizing minimum possible, over distribution with the same average and standard deviation, expected profit, according to some criteria. We discuss the practical suitability of the objective function set and the solution derived, for the Newsvendor Problem, and other such random optimization problems.

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