The projected Barzilai–Borwein method with fall-back for strictly convex QCQP problems with separable constraints

Pospíšil, Lukáš; Dostál, Zdeněk

doi:10.1016/j.matcom.2017.10.003

Cited by 10 publications

(8 citation statements)

References 16 publications

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“…Preliminary numerical results suggest that the proposed rule shows benefits also in gradient projection methods for general box-constrained non-quadratic optimization problems. Future work will concern the generalization of this analysis to the cases of more general constraints or more complex schemes like the scaled gradient projection methods [26,38,22].…”

Section: Discussionmentioning

confidence: 99%

“…For what concerns the case of the constrained optimization problems, the role of the steplength rules is not so deeply investigated. Many well-known gradient projection methods for constrained optimization simply exploit the same steplength selections designed for the unconstrained case in combination with some kind of line search strategy [12,13,14,15,16,17,18,19,20,21,22]. However, most of the steplength selections for unconstrained problems are designed for achieving a fast annihilation of the gradient of the objective function, that is not the main goal in constrained optimization.…”

Section: Introductionmentioning

confidence: 99%

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Steplength selection in gradient projection methods for box-constrained quadratic programs

Crisci

Ruggiero

Zanni

2019

Applied Mathematics and Computation

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The role of the steplength selection strategies in gradient methods has been widely investigated in the last decades. Starting from the work of Barzilai and Borwein (1988), many efficient steplength rules have been designed, that contributed to make the gradient approaches an effective tool for the large-scale optimization problems arising in important real-world applications. Most of these steplength rules have been thought in unconstrained optimization, with the aim of exploiting some second-order information for achieving a fast annihilation of the gradient of the objective function. However, these rules are successfully used also within gradient projection methods for constrained optimization, though, to our knowledge, a detailed analysis of the effects of the constraints on the steplength selections is still not available. In this work we investigate how the presence of the box constraints affects the spectral properties of the Barzilai-Borwein rules in quadratic programming problems. The proposed analysis suggests the introduction of new steplength selection strategies specifically designed for taking account of the active constraints at each iteration. The results of a set of numerical experiments show the effectiveness of the new rules with respect to other state of the art steplength selections and their potential usefulness also in case of box-constrained non-quadratic optimization problems.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Steplength selection in gradient projection methods for box-constrained quadratic programs

Crisci

Ruggiero

Zanni

2019

Applied Mathematics and Computation

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“…with a step-lengthᾱ ∈ (0, A −1 ). Decrease of the function value for a convex QP on a general closed convex set has been proven in (6), (11).…”

Section: The Solution Of γ Subproblemmentioning

confidence: 99%

Low-cost scalable discretization, prediction, and feature selection for complex systems

et al. 2020

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Finding reliable discrete approximations of complex systems is a key prerequisite when applying many of the most popular modeling tools. Common discretization approaches (e.g., the very popular K-means clustering) are crucially limited in terms of quality, parallelizability, and cost. We introduce a low-cost improved quality scalable probabilistic approximation (SPA) algorithm, allowing for simultaneous data-driven optimal discretization, feature selection, and prediction. We prove its optimality, parallel efficiency, and a linear scalability of iteration cost. Cross-validated applications of SPA to a range of large realistic data classification and prediction problems reveal marked cost and performance improvements. For example, SPA allows the data-driven next-day predictions of resimulated surface temperatures for Europe with the mean prediction error of 0.75°C on a common PC (being around 40% better in terms of errors and five to six orders of magnitude cheaper than with common computational instruments used by the weather services).

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“…Another approach is to use Interior-Point (IP, [25]) method-we eliminate the inequality constraints using barrier functions and solve the sequence of resulting saddle-point problems. In this paper, we examine the projected gradient descent method [26][27][28], especially SPG-QP (Spectral Projected Gradients [29] for QP [5]).…”

Section: Introductionmentioning

confidence: 99%

Non-Monotone Projected Gradient Method in Linear Elasticity Contact Problems with Given Friction

et al. 2020

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We are focusing on the algorithms for solving the large-scale convex optimization problem in linear elasticity contact problems discretized by Finite Element method (FEM). The unknowns of the problem are the displacements of the FEM nodes, the corresponding objective function is defined as a convex quadratic function with symmetric positive definite stiffness matrix and additional non-linear term representing the friction in contact. The feasible set constraints the displacement subject to non-penetration conditions. The dual formulation of this optimization problem is well-known as a Quadratic Programming (QP) problem and can be considered as a most basic non-linear optimization problem. Understanding these problems and the development of efficient algorithms for solving them play the crucial role in the large-scale problems in practical applications. We shortly review the theory and examine the behavior and the efficiency of Spectral Projected Gradient method modified for QP problems (SPG-QP) on the solution of a toy example in MATLAB environment.

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The projected Barzilai–Borwein method with fall-back for strictly convex QCQP problems with separable constraints

Cited by 10 publications

References 16 publications

Steplength selection in gradient projection methods for box-constrained quadratic programs

Steplength selection in gradient projection methods for box-constrained quadratic programs

Low-cost scalable discretization, prediction, and feature selection for complex systems

Non-Monotone Projected Gradient Method in Linear Elasticity Contact Problems with Given Friction

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