## The U-Lagrangian of a convex function

Venue: | Trans. Amer. Math. Soc |

Citations: | 26 - 7 self |

### BibTeX

@ARTICLE{Lemaréchal_theu-lagrangian,

author = {Claude Lemaréchal and François Oustry and Claudia Sagastizábal},

title = {The U-Lagrangian of a convex function},

journal = {Trans. Amer. Math. Soc},

year = {},

pages = {711--729}

}

### OpenURL

### Abstract

Abstract. At a given point p, a convex function f is differentiable in a certain subspace U (the subspace along which ∂f(p) has 0-breadth). This property opens the way to defining a suitably restricted second derivative of f at p. We do this via an intermediate function, convex on U. We call this function the U-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization. 1.

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Citation Context ...Yosida regularization. Indeed, the present paper clarifies and formalizes the theory sketched in §3.2 of [15]; for a related subject see also [29], [25]. Our notation follows closely that of [28] and =-=[11]-=-. The space Rn is equipped with a scalar product 〈·, ·〉, and‖·‖ is the associated norm; in a subspace S, we will write 〈·, ·〉S and ‖·‖S for the induced scalar product and norm. The open ball of Rn cen... |

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Citation Context ...ple the case where f =maxifiwith smooth fi’s; then a minimization algorithm of the SQP-type will converge superlinearly, even if the second-order behaviour of f is identified in the ridge only ([26], =-=[6]-=-). Here, starting from results presented in [14] and [15], we take advantage of these observations. After some preliminary theory in §2, we define our key-objects in §3: the U-Lagrangian and its deriv... |

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Citation Context ...thms. Finally, we establish a connection with the Moreau-Yosida regularization. 1. Introduction This paper deals with higher-order expansions of a nonsmooth function, a problem addressed in [4], [5], =-=[7]-=-, [9], [13], [25], and [31] among others. The initial motivation for our present work lies in the following facts. When trying to generalize the classical second-order Taylor expansion of a function f... |

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Citation Context ...ween sets, say ∂f(p + h) − ∂f(p) (1.1) . ‖h‖ Giving a sensible meaning to the minus-sign in this expression is a difficult problem, to say the least; it has received only abstract answers so far; see =-=[1]-=-, [3], [10], [12], [16], [18], [23], [24], [30]. However, here are two crucial observations (already mentioned in [22]): – There is a subspace U (the “ridge”) in which the first-order approximation f ... |

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Citation Context ...lly, we establish a connection with the Moreau-Yosida regularization. 1. Introduction This paper deals with higher-order expansions of a nonsmooth function, a problem addressed in [4], [5], [7], [9], =-=[13]-=-, [25], and [31] among others. The initial motivation for our present work lies in the following facts. When trying to generalize the classical second-order Taylor expansion of a function f at a nondi... |

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Citation Context ... which we compute first the V-component of the increment p+ − p, and then its U-component. This idea of decomposing the move from p to p+ in a “vertical” and a “horizontal” step can be traced back to =-=[8]-=-. Algorithm 4.5. V-Step. Compute a solution δv ∈Vof (4.8) min{f(p +0⊕δv) :δv ∈V} and set p ′ := p +0⊕δv. U-Step. Make a Newton step in p ′ + U: compute the solution δu ∈U of g ′ (4.9) U +HUf(p)δu =0, ... |

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Citation Context ...e advantage of these observations. After some preliminary theory in §2, we define our key-objects in §3: the U-Lagrangian and its derivatives. In §4 we give some specific examples (further studied in =-=[17]-=-, [20]): how the U-Lagrangian specializes in an NLP and an SDP Received by the editors July 18, 1996 and, in revised form, August 1, 1997. 1991 Mathematics Subject Classification. Primary 49J52, 58C20... |

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Citation Context ...ion algorithms. Finally, we establish a connection with the Moreau-Yosida regularization. 1. Introduction This paper deals with higher-order expansions of a nonsmooth function, a problem addressed in =-=[4]-=-, [5], [7], [9], [13], [25], and [31] among others. The initial motivation for our present work lies in the following facts. When trying to generalize the classical second-order Taylor expansion of a ... |

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Citation Context ... Finally, we establish a connection with the Moreau-Yosida regularization. 1. Introduction This paper deals with higher-order expansions of a nonsmooth function, a problem addressed in [4], [5], [7], =-=[9]-=-, [13], [25], and [31] among others. The initial motivation for our present work lies in the following facts. When trying to generalize the classical second-order Taylor expansion of a function f at a... |

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Citation Context ...hen a minimization algorithm of the SQP-type will converge superlinearly, even if the second-order behaviour of f is identified in the ridge only ([26], [6]). Here, starting from results presented in =-=[14]-=- and [15], we take advantage of these observations. After some preliminary theory in §2, we define our key-objects in §3: the U-Lagrangian and its derivatives. In §4 we give some specific examples (fu... |

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Citation Context ...l” strong convexity of f ∗ at g, say, f ∗ (g + s) ≥ f ∗ (g)+〈s, p〉 + 1 2 c‖s‖2 + o(‖s‖ 2 ) for some c>0. However, the above inequality is hopeless for an s of the form s =0⊕v(see §4 in [14]; see also =-=[2]-=- for related developments). To obtain radial strong convexity on V, we introduce the function f ∗ (g + s)+ 1 2 c‖sV‖ 2 (3.3) V. License or copyright restrictions may apply to redistribution; see http:... |

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Citation Context ...lgorithms. Finally, we establish a connection with the Moreau-Yosida regularization. 1. Introduction This paper deals with higher-order expansions of a nonsmooth function, a problem addressed in [4], =-=[5]-=-, [7], [9], [13], [25], and [31] among others. The initial motivation for our present work lies in the following facts. When trying to generalize the classical second-order Taylor expansion of a funct... |

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