Fast Gradient Descent for Convex Minimization Problems with an Oracle Producing a (δ, L)-Model of Function at the Requested Point

Gasnikov, A. V.; Tyurin, Alexander

doi:10.1134/s0965542519070078

Cited by 14 publications

(11 citation statements)

References 14 publications

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“…It it shown in [31] that in order to construct the classical Frank-Wolfe method instead of an auxiliary problem φ k+1 (x) = α k+1 ψ δk (x, y k+1 ) + V [u k ](x) in Algorithm 2 for m = 0 and µ = 0 (see also section 3, [31]) we can take an auxiliary problem φ k+1 (x) = α k+1 ψ δk (x, y k+1 ). Let us look at this substitution from the view of δ k -precision from Definition 2.11.…”

Section: Universal Conditional Gradient (Frank-wolfe) Methodsmentioning

confidence: 99%

“…Note that in Definition 2.1 we allow L to depend on δ. Definition 2.1 is a generalization of (δ, L)-model from [29,31,65], where µ = 0 and m = 0. Further, we denote (δ, L, 0, 0, V )-model as (δ, L)-model.…”

Section: Inexact Model In Minimization Problems Definitions and Examplesmentioning

confidence: 99%

“…The resulting model includes as a particular case inexact oracle model and relative smoothness framework, and allows to obtain many optimization methods as a particular case, including conditional gradient method [26], Bregman proximal gradient method [11] and its application to optimal transport [69] and Wasserstein barycenter [65] problems, general Catalyst acceleration technique [44], (accelerated) composite gradient methods [7,52], (accelerated) level methods [42,50]. First attempts to propose this generalization were made in [29,65] for nonaccelerated methods and in [31] for accelerated methods, yet without relative smoothness paradigm. In this paper we propose the inexact model in a very general setting including adaptivity of the algorithms to the parameter L, possible relative strong convexity and relative smoothness.…”

Section: Introductionmentioning

confidence: 99%

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Inexact Relative Smoothness and Strong Convexity for Optimization and Variational Inequalities by Inexact Model

Stonyakin¹,

Tyurin²,

Gasnikov³

et al. 2020

Preprint

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In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities.

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Section: Universal Conditional Gradient (Frank-wolfe) Methodsmentioning

confidence: 99%

Section: Inexact Model In Minimization Problems Definitions and Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Inexact Relative Smoothness and Strong Convexity for Optimization and Variational Inequalities by Inexact Model

Stonyakin¹,

Tyurin²,

Gasnikov³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…For the weight ratio assignment of multi-objective weighting function, the weight ratio assignment in this paper is shown in formula (23).…”

Section: Control Strategy and Optimization Of Hessmentioning

confidence: 99%

“…The traditional method for global optimization is a numerical optimization method based on gradient information, which is discussed in reference. [23][24][25][26] The optimization idea of the gradient descent method is to use the negative gradient direction of the current position as the search direction. The closer the gradient descent method is to the target value, the smaller the step size and the slower the advance.…”

Section: Control Strategy and Optimization Of Hessmentioning

confidence: 99%

A new adaptive PSO-PID control strategy of hybrid energy storage system for electric vehicles

2014

Advances in Mechanical Engineering

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Research on optimization of control strategy for hybrid energy storage system (HESS) of the electric vehicle (EV), a new adaptive control strategy based on particle swarm optimization (PSO) algorithm is proposed in this paper. The steady-state power of the filtered power is used as the ideal output power of the battery. For the steady-state current output of the battery, the output power of the ultracapacitor is dynamically adjusted by the proportional-integral-derivative (PID) controller to construct a power difference control structure. The parameters of PID controller are optimized by PSO algorithm, and the target test is compared and analyzed based on MATLAB/Advisor. The research results show that the proposed PSO-PID control strategy can quickly eliminate the power deviation and achieve the approximate global optimization of the EV energy management strategy. Compared with the pre-optimized PID control strategy, the output current and power of the battery pack are smoother and the total power consumption is reduced by 3.8360% and 0.5125%, respectively. Then, the energy consumption parameters of PSO-PID are compared with the theoretical minimum energy consumption calculated by dynamic programming (DP) algorithm, and the deviation is less than 1% under both conditions.

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Gradient Methods for Problems with Inexact Model of the Objective

Stonyakin

Dvinskikh

Dvurechensky

et al. 2019

Lecture Notes in Computer Science

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We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes inexact oracle [19] and relative smoothness condition [43]. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [49]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments.

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Fast Gradient Descent for Convex Minimization Problems with an Oracle Producing a (δ, L)-Model of Function at the Requested Point

Cited by 14 publications

References 14 publications

Inexact Relative Smoothness and Strong Convexity for Optimization and Variational Inequalities by Inexact Model

Inexact Relative Smoothness and Strong Convexity for Optimization and Variational Inequalities by Inexact Model

A new adaptive PSO-PID control strategy of hybrid energy storage system for electric vehicles

Gradient Methods for Problems with Inexact Model of the Objective

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