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Applications of convex analysis within mathematics Aragón Artacho, Francisco Javier ; ; et al in Mathematical Programming (in press) In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of ... [more ▼] In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis. [less ▲] Detailed reference viewed: 232 (24 UL)Recent Results on Douglas–Rachford Methods for Combinatorial Optimization Problems Aragón Artacho, Francisco Javier ; ; in Journal of Optimization Theory and Applications (in press) We discuss recent positive experiences applying convex feasibility algorithms of Douglas-Rachford type to highly combinatorial and far from convex problems. Detailed reference viewed: 155 (16 UL)Globally convergent algorithms for finding zeros of duplomonotone mappings Aragón Artacho, Francisco Javier ; Fleming, Ronan MT in Optimization Letters (2015), 3(3), 569584 We introduce a new class of mappings, called duplomonotone, which is strictly broader than the class of monotone mappings. We study some of the main properties of duplomonotone functions and provide ... [more ▼] We introduce a new class of mappings, called duplomonotone, which is strictly broader than the class of monotone mappings. We study some of the main properties of duplomonotone functions and provide various examples, including nonlinear duplomonotone functions arising from the study of systems of biochemical reactions. Finally, we present three variations of a derivative-free line search algorithm for finding zeros of systems of duplomonotone equations, and we prove their linear convergence to a zero of the function. [less ▲] Detailed reference viewed: 329 (27 UL)Accelerating the DC algorithm for smooth functions Aragón Artacho, Francisco Javier ; Fleming, Ronan MT ; Phan, Vuong E-print/Working paper (2015) We introduce two new algorithms to minimise smooth difference of convex (DC) functions that accelerate the convergence of the classical DC algorithm (DCA). We prove that the point computed by DCA can be ... [more ▼] We introduce two new algorithms to minimise smooth difference of convex (DC) functions that accelerate the convergence of the classical DC algorithm (DCA). We prove that the point computed by DCA can be used to define a descent direction for the objective function evaluated at this point. Our algorithms are based on a combination of DCA together with a line search step that uses this descent direction. Convergence of the algorithms is proved and the rate of convergence is analysed under the Łojasiewicz property of the objective function. We apply our algorithms to a class of smooth DC programs arising in the study of biochemical reaction networks, where the objective function is real analytic and thus satisfies the Łojasiewicz property. Numerical tests on various biochemical models clearly show that our algorithms outperforms DCA, being on average more than four times faster in both computational time and the number of iterations. The algorithms are globally convergent to a non-equilibrium steady state of a biochemical network, with only chemically consistent restrictions on the network topology. [less ▲] Detailed reference viewed: 172 (16 UL)Douglas-Rachford Feasibility Methods for Matrix Completion Problems Aragón Artacho, Francisco Javier ; ; in ANZIAM Journal (2014), 55(4), 299-326 In this paper we give general recommendations for successful application of the Douglas-Rachford reflection method to convex and non-convex real matrix-completion problems. These guidelines are ... [more ▼] In this paper we give general recommendations for successful application of the Douglas-Rachford reflection method to convex and non-convex real matrix-completion problems. These guidelines are demonstrated by various illustrative examples. [less ▲] Detailed reference viewed: 116 (1 UL)Metric subregularity of the convex subdifferential in Banach spaces Aragón Artacho, Francisco Javier ; in Journal of Nonlinear and Convex Analysis (2014), 15(1), 35-47 In [2] we characterized in terms of a quadratic growth condition various metric regularity properties of the subdifferential of a lower semicontinuous convex function acting in a Hilbert space. Motivated ... [more ▼] In [2] we characterized in terms of a quadratic growth condition various metric regularity properties of the subdifferential of a lower semicontinuous convex function acting in a Hilbert space. Motivated by some recent results in [16] where the authors extend to Banach spaces the characterization of the strong regularity, we extend as well the characterizations for the metric subregularity and the strong subregularity given in [2] to Banach spaces. We also notice that at least one implication in these characterizations remains valid for the limiting subdifferential without assuming convexity of the function in Asplund spaces. Additionally, we show some direct implications of the characterizations for the convergence of the proximal point algorithm, and we provide some characterizations of the metric subregularity and calmness properties of solution maps to parametric generalized equations [less ▲] Detailed reference viewed: 53 (7 UL)Local convergence of quasi-Newton methods under metric regularity Aragón Artacho, Francisco Javier ; ; et al in Computational Optimization and Applications (2014), 58(1), 225-247 We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden ... [more ▼] We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results. [less ▲] Detailed reference viewed: 110 (12 UL)Walking on Real Numbers Aragón Artacho, Francisco Javier ; ; et al in Mathematical Intelligencer (2013), 35(1), 42-60 Motivated by the desire to visualize large mathematical data sets, especially in number theory, we offer various tools for representing ﬂoating point numbers as planar(or three dimensional) walks and for ... [more ▼] Motivated by the desire to visualize large mathematical data sets, especially in number theory, we offer various tools for representing ﬂoating point numbers as planar(or three dimensional) walks and for quantitatively measuring their “randomness.” [less ▲] Detailed reference viewed: 51 (5 UL)Global convergence of a non-convex Douglas-Rachford iteration Aragón Artacho, Francisco Javier ; in Journal of Global Optimization (2013), 57(3), 753-769 We establish a region of convergence for the proto-typical non-convex Douglas–Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration ... [more ▼] We establish a region of convergence for the proto-typical non-convex Douglas–Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration Borwein and Sims (Fixed-point algorithms for inverse problems in science and engineering, pp. 93–109, 2011) was only able to establish local convergence, and was ineffective in that no explicit region of convergence could be given. [less ▲] Detailed reference viewed: 101 (10 UL)Recent Results on Douglas–Rachford Methods Aragón Artacho, Francisco Javier ; ; in Serdica Mathematical Journal (2013), 39 Recent positive experiences applying convex feasibility algorithms of Douglas–Rachford type to highly combinatorial and far from convex problems are described. Detailed reference viewed: 46 (6 UL)A Lyusternik - Graves theorem for the proximal point method Aragón Artacho, Francisco Javier ; in Computational Optimization and Applications (2012), 52(3), 785-803 We consider a generalized version of the proximal point algorithm for solving the perturbed inclusion y∈T(x), where y is a perturbation element near 0 and T is a set-valued mapping acting from a Banach ... [more ▼] We consider a generalized version of the proximal point algorithm for solving the perturbed inclusion y∈T(x), where y is a perturbation element near 0 and T is a set-valued mapping acting from a Banach space X to a Banach space Y which is metrically regular around some point (x̅,0) in its graph. We study the behavior of the convergent iterates generated by the algorithm and we prove that they inherit the regularity properties of T, and vice versa. We analyze the cases when the mapping T is metrically regular and strongly regular. [less ▲] Detailed reference viewed: 90 (5 UL)Enhanced metric regularity and Lipschitzian properties of variational systems Aragón Artacho, Francisco Javier ; in Journal of Global Optimization (2011), 50(1), 145-167 This paper mainly concerns the study of a large class of variational systems governed by parametric generalized equations, which encompass variational and hemivariational inequalities, complementarity ... [more ▼] This paper mainly concerns the study of a large class of variational systems governed by parametric generalized equations, which encompass variational and hemivariational inequalities, complementarity problems, first-order optimality conditions, and other optimization-related models important for optimization theory and applications. An efficient approach to these issues has been developed in our preceding work (Aragón Artacho and Mordukhovich in Nonlinear Anal 72:1149–1170, 2010) establishing qualitative and quantitative relationships between conventional metric regularity/subregularity and Lipschitzian/calmness properties in the framework of parametric generalized equations in arbitrary Banach spaces. This paper provides, on one hand, significant extensions of the major results in op.cit. to partial metric regularity and to the new hemiregularity property. On the other hand, we establish enhanced relationships between certain strong counterparts of metric regularity/hemiregularity and single-valued Lipschitzian localizations. The results obtained are new in both finite-dimensional and infinite-dimensional settings. [less ▲] Detailed reference viewed: 87 (9 UL)Metric regularity of Newton's iteration Aragón Artacho, Francisco Javier ; ; et al in SIAM Journal on Control & Optimization (2011), 49(2), 339-362 For a version of Newton's method applied to a generalized equation with a parameter, we extend the paradigm of the Lyusternik–Graves theorem to the framework of a mapping acting from the pair “parameter ... [more ▼] For a version of Newton's method applied to a generalized equation with a parameter, we extend the paradigm of the Lyusternik–Graves theorem to the framework of a mapping acting from the pair “parameter-starting point” to the set of corresponding convergent Newton sequences. Under ample parameterization, metric regularity of the mapping associated with convergent Newton sequences becomes equivalent to the metric regularity of the mapping associated with the generalized equation. We also discuss an inexact Newton method and present an application to discretized optimal control. [less ▲] Detailed reference viewed: 90 (7 UL)Metric regularity and Lipschitzian stability of parametric variational systems Aragón Artacho, Francisco Javier ; in Nonlinear Analysis: Theory, Methods and Applications (2010), 72(3-4), 1149-1170 The paper concerns the study of variational systems described by parameterized generalized equations/variational conditions important for many aspects of nonlinear analysis, optimization, and their ... [more ▼] The paper concerns the study of variational systems described by parameterized generalized equations/variational conditions important for many aspects of nonlinear analysis, optimization, and their applications. Focusing on the fundamental properties of metric regularity and Lipschitzian stability, we establish various qualitative and quantitative relationships between these properties for multivalued parts/fields of parametric generalized equations and the corresponding solution maps for them in the framework of arbitrary Banach spaces of decision and parameter variables. [less ▲] Detailed reference viewed: 97 (6 UL)Characterization of metric regularity of subdifferentials Aragón Artacho, Francisco Javier ; in Journal of Convex Analysis (2008), 15(2), 365-380 We study regularity properties of the subdifferential of proper lower semicontinuous convex functions in Hilbert spaces. More precisely, we investigate the metric regularity and subregularity, the strong ... [more ▼] We study regularity properties of the subdifferential of proper lower semicontinuous convex functions in Hilbert spaces. More precisely, we investigate the metric regularity and subregularity, the strong regularity and subregularity of such a subdifferential. We characterize each of these properties in terms of a growth condition involving the function. [less ▲] Detailed reference viewed: 148 (6 UL)On the inner and outer norms of sublinear mappings Aragón Artacho, Francisco Javier ; in Set-Valued Analysis (2007), 15(1), 61-65 In this short note we show that the outer norm of a sublinear mapping F, acting between Banach spaces X and Y and with dom F = X, is finite only if F is single-valued. This implies in particular that for ... [more ▼] In this short note we show that the outer norm of a sublinear mapping F, acting between Banach spaces X and Y and with dom F = X, is finite only if F is single-valued. This implies in particular that for a sublinear multivalued mapping the inner and the outer norms cannot be finite simultaneously. [less ▲] Detailed reference viewed: 85 (2 UL)A new and self-contained proof of Borwein's norm duality theorem Aragón Artacho, Francisco Javier in Set-Valued Analysis (2007), 15(3), 307-315 Borwein’s norm duality theorem establishes the equality between the outer (inner) norm of a sublinear mapping and the inner (outer) norm of its adjoint mappings. In this note we provide an extended ... [more ▼] Borwein’s norm duality theorem establishes the equality between the outer (inner) norm of a sublinear mapping and the inner (outer) norm of its adjoint mappings. In this note we provide an extended version of this theorem with a new and self-contained proof relying only on the Hahn-Banach theorem. We also give examples showing that the assumptions of the theorem cannot be relaxed. [less ▲] Detailed reference viewed: 101 (3 UL)Convergence of the proximal point method for metrically regular mappings Aragón Artacho, Francisco Javier in ESAIM: Proceedings and Surveys (2007), 17 In this paper we consider the following general version of the proximal point algorithm for solving the inclusion T(x) ∋ 0, where T is a set-valued mapping acting from a Banach space X to a Banach space Y ... [more ▼] In this paper we consider the following general version of the proximal point algorithm for solving the inclusion T(x) ∋ 0, where T is a set-valued mapping acting from a Banach space X to a Banach space Y. First, choose any sequence of functions gn : X → Y with gn(0) = 0 that are Lipschitz continuous in a neighborhood of the origin. Then pick an initial guess x0 and find a sequence xn by applying the iteration gn(xn1-xn)+T(xn+1) ∋ 0 for n = 0,1,... We prove that if the Lipschitz constants of gn are bounded by half the reciprocal of the modulus of regularity of T, then there exists a neighborhood O of x̅ (x̅ being a solution to T(x) ∋ 0) such that for each initial point x₀ ∈ O one can find a sequence xn generated by the algorithm which is linearly convergent to x̅. Moreover, if the functions gn have their Lipschitz constants convergent to zero, then there exists a sequence starting from x₀ ∈ O which is superlinearly convergent to x̅. Similar convergence results are obtained for the cases when the mapping T is strongly subregular and strongly regular. [less ▲] Detailed reference viewed: 136 (8 UL)Uniformity and inexact version of a proximal method for metrically regular mappings Aragón Artacho, Francisco Javier ; in Journal of Mathematical Analysis and Applications (2007), 335(1), 168-183 We study stability properties of a proximal point algorithm for solving the inclusion 0∈T(x) when T is a set-valued mapping that is not necessarily monotone. More precisely we show that the convergence of ... [more ▼] We study stability properties of a proximal point algorithm for solving the inclusion 0∈T(x) when T is a set-valued mapping that is not necessarily monotone. More precisely we show that the convergence of our algorithm is uniform, in the sense that it is stable under small perturbations whenever the set-valued mapping T is metrically regular at a given solution. We present also an inexact proximal point method for strongly metrically subregular mappings and show that it is super-linearly convergent to a solution to the inclusion 0∈T(x). [less ▲] Detailed reference viewed: 88 (5 UL) |
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