Vo Phat

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Niall Meehan

Griffith College Dublin

Eva Botella-Ordinas

Universidad Autónoma de Madrid

Fabio Cuzzolin

Oxford Brookes University

Miguel Bagajewicz

University of Oklahoma

Harshan Kumarasingham

University of Edinburgh

Florentin Smarandache

University of New Mexico

Angelamaria Cardone

University of Salerno

huyen tran

Ho Chi Minh City University of technology

Nasreen Kausar

Yildiz Technical University

Olga Dror

Texas A&M University

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Papers by Vo Phat

$Research paper thumbnail of On Second-order Conditions for Quasiconvexity and Pseudoconvexity of $\mathcal{C}^{1,1}$-smooth Functions$

On Second-order Conditions for Quasiconvexity and Pseudoconvexity of $\mathcal{C}^{1,1}$-smooth Functions

For a C 2-smooth function on a finite-dimensional space, a necessary condition for its quasiconve... more For a C 2-smooth function on a finite-dimensional space, a necessary condition for its quasiconvexity is the positive semidefiniteness of its Hessian matrix on the subspace orthogonal to its gradient, whereas a sufficient condition for its strict pseudoconvexity is the positive definiteness of its Hessian matrix on the subspace orthogonal to its gradient. Our aim in this paper is to extend those conditions for C 1,1-smooth functions by using the Fréchet and Mordukhovich second-order subdifferentials.

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Variational Convexity of Functions and Variational Sufficiency in Optimization

The paper is devoted to the study, characterizations, and applications of variational convexity o... more The paper is devoted to the study, characterizations, and applications of variational convexity of functions, the property that has been recently introduced by Rockafellar together with its strong counterpart. First we show that these variational properties of an extended-real-valued function are equivalent to, respectively, the conventional (local) convexity and strong convexity of its Moreau envelope. Then we derive new characterizations of both variational convexity and variational strong convexity of general functions via their second-order subdifferentials (generalized Hessians), which are coderivatives of subgradient mappings. We also study relationships of these notions with local minimizers and tilt-stable local minimizers. The obtained results are used for characterizing related notions of variational and strong variational sufficiency in composite optimization with applications to nonlinear programming.

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Generalized Damped Newton Algorithms in Nonsmooth Optimization via Second-Order Subdifferentials

The paper proposes and develops new globally convergent algorithms of the generalized damped Newt... more The paper proposes and develops new globally convergent algorithms of the generalized damped Newton type for solving important classes of nonsmooth optimization problems. These algorithms are based on the theory and calculations of second-order subdifferentials of nonsmooth functions with employing the machinery of second-order variational analysis and generalized differentiation. First we develop a globally superlinearly convergent damped Newton-type algorithm for the class of continuously differentiable functions with Lipschitzian gradients, which are nonsmooth of second order. Then we design such a globally convergent algorithm to solve a structured class of nonsmooth quadratic composite problems with extended-real-valued cost functions, which typically arise in machine learning and statistics. Finally, we present the results of numerical experiments and compare the performance of our main algorithm applied to an important class of Lasso problems with those achieved by other first-order and second-order optimization algorithms.

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A Generalized Newton Method for Subgradient Systems

This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions ... more This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order subdifferential of such functions that enjoys extensive calculus rules and can be efficiently computed for broad classes of extended-real-valued functions. Based on this and on metric regularity and subregularity properties of subgradient mappings, we establish verifiable conditions ensuring well-posedness of the proposed algorithm and its local superlinear convergence. The obtained results are also new for the class of equations defined by continuously differentiable functions with Lipschitzian gradients (C 1,1 functions), which is the underlying case of our consideration. The developed algorithms for prox-regular functions and its extension to a structured class of composite functions are formulated in terms of proximal mappings and forward-backward envelopes. Besides numerous illustrative examples and comparison with known algorithms for C 1,1 functions and generalized equations, the paper presents applications of the proposed algorithms to regularized least square problems arising in statistics, machine learning, and related disciplines

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Globally Convergent Coderivative-Based Generalized Newton Methods in Nonsmooth Optimization

This paper proposes and justifies two new globally convergent Newton-type methods to solve uncons... more This paper proposes and justifies two new globally convergent Newton-type methods to solve unconstrained and constrained problems of nonsmooth optimization by using tools of variational analysis and generalized differentiation. Both methods are coderivative-based and employ generalized Hessians (coderivatives of subgradient mappings) associated with objective functions, which are either of class C1,1, or are represented in the form of convex composite optimization, where one of the terms may be extended-real-valued. The proposed globally convergent algorithms are of two types. The first one extends the damped Newton method and requires positive-definiteness of the generalized Hessians for its well-posedness and efficient performance, while the other algorithm is of the Levenberg-Marquardt type being well-defined when the generalized Hessians are merely positive-semidefinite. The obtained convergence rates for both methods are at least linear, but becomes superlinear under the so-cal...

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Second-order characterizations of quasiconvexity and pseudoconvexity for differentiable functions with Lipschitzian derivatives

Optimization Letters, 2020

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Second-order characterizations of C1-smooth robustly quasiconvex functions

Operations Research Letters, 2018

Abstract This paper investigates the possibility of using the Frechet and Mordukhovich second-ord... more Abstract This paper investigates the possibility of using the Frechet and Mordukhovich second-order subdifferentials to characterize the robust quasiconvexity of C 1 -smooth functions. We set up a necessary condition for the robust quasiconvexity of C 1 , 1 -smooth functions and univariate C 1 -smooth ones. We also show that the established necessary condition is indeed a sufficient one for the robust quasiconvexity of C 1 -smooth functions.

$Research paper thumbnail of On Second-order Conditions for Quasiconvexity and Pseudoconvexity of $\mathcal{C}^{1,1}$-smooth Functions$

On Second-order Conditions for Quasiconvexity and Pseudoconvexity of $\mathcal{C}^{1,1}$-smooth Functions

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Variational Convexity of Functions and Variational Sufficiency in Optimization

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Generalized Damped Newton Algorithms in Nonsmooth Optimization via Second-Order Subdifferentials

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A Generalized Newton Method for Subgradient Systems

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Globally Convergent Coderivative-Based Generalized Newton Methods in Nonsmooth Optimization

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Second-order characterizations of quasiconvexity and pseudoconvexity for differentiable functions with Lipschitzian derivatives