## A quasi-Newton penalty barrier method for convex minimization problems (2002)

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Citations: | 5 - 0 self |

### BibTeX

@MISC{Armand02aquasi-newton,

author = {Paul Armand and Paul Armand},

title = {A quasi-Newton penalty barrier method for convex minimization problems},

year = {2002}

}

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### Abstract

We describe an infeasible interior point algorithm for convex minimization problems. The method uses quasi-Newton techniques for approximating the second derivatives and providing superlinear convergence. We propose a new feasibility control of the iterates by introducing shift variables and by penalizing them in the barrier problem. We prove global convergence under standard conditions on the problem data, without any assumption on the behavior of the algorithm.

### Citations

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(Show Context)
Citation Context ...d the constraints satisfy some qualification assumption, the Karush-Kuhn-Tucker (KKT) optimality conditions of problem (1.1) can i=1A QUASI-NEWTON PENALTY BARRIER METHOD 9 be written as follows (see =-=[12]-=- for example): if x is a solution of (1.1) then there exists a vector of multipliers λ ∈ Rm such that ⎧ ∇ f (x) +∇c(x)λ = 0, ⎪⎨ c(x) ≤ 0, (2.2) C(x)λ = 0, ⎪⎩ λ ≥ 0, where C(x) is the diagonal matrix d... |

462 | Primal-Dual Interior-Point Methods - Wright - 1997 |

431 |
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(Show Context)
Citation Context ...associated ℓ2 norm. A function is of class C 1,1 if it is continuously differentiable and if its derivative is Lipschitz continuous. Let us recall some definitions of convex analysis (see for example =-=[7]-=-). A function φ : R n → R is strongly convex with modulus κ>0, if the function φ(·)− κ 2 ‖·‖2 is convex. For a differentiable function, the strong convexity is equivalent to the strong monotonicity of... |

314 | Nonlinear Programming: Sequential Unconstrained Minimization Techniques - Fiacco, McCormick - 1990 |

105 | A trust region method based on interior point techniques for nonlinear programming
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- 2000
(Show Context)
Citation Context ...f the transformation was to preserve the convexity of the original problem, that would not have been the case with the use of slack variables, that is with the transformation c(x) + s = 0, s ≥ 0 (see =-=[3]-=-, for example). The barrier problem associated to (1.4) is the following equality constraint problem: ⎧ ⎪⎨ m∑ min φµ(x) := f (x) − µ log(si − ci(x)), ⎪⎩ i=1 s = 0. In this approach, the inequality s −... |

46 |
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Citation Context ...). By definition of the sets B and N, we obtain finally ¯λ ⊤ j N wN − ( λ j ) ⊤cB(¯x) B + (σ − ¯λ) ⊤ s j ≤ 2mµ j + o(µ j ). (4.9)22 ARMAND The remainder of the proof uses an argument due to McLinden =-=[10]-=-. Let us define Ɣ j := � j w j − µ j e and � j := S j (σ − λ j ) − µ j e. One has for all indices i: w j i = µ j + Ɣ j i λ j , λ i j i = µ j + Ɣ j i w j i Substituting this in (4.9) gives and s j i = ... |

43 |
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(Show Context)
Citation Context ... L1. The convergence analysis rests on two lemmas. The first one is a consequence of the sufficient decrease condition (3.5). The second one is a property of the BFGS update formula (3.6). Lemma B.1 (=-=[4]-=-). Suppose that ψσ,µ is C1,1 on an open convex neighborhood N of the level set L1. There exists a constant K3 > 0 such that for any z ∈ L1 and for any descent direction d of ψσ,µ, if the step length α... |

30 | A primal-dual interior-point method for nonlinear programming with strong global and local convergence properties
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- 2003
(Show Context)
Citation Context ... in methods that solve inequality constraints problems by generating feasible iterates. This scheme has been used in some feasible directions algorithms, see [6, 8] and the recent work of Tits et al. =-=[15]-=-. It consists of keeping the iterates on one side of each equality constraint and penalizing the iterates to force them to go more and more near the boundary of these fictive inequality constraints. T... |

16 | On the convergence of the variable metric algorithm - POWELL - 1971 |

14 |
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(Show Context)
Citation Context ...scheme to incorporate equality constraints in methods that solve inequality constraints problems by generating feasible iterates. This scheme has been used in some feasible directions algorithms, see =-=[6, 8]-=- and the recent work of Tits et al. [15]. It consists of keeping the iterates on one side of each equality constraint and penalizing the iterates to force them to go more and more near the boundary of... |

13 | A feasible BFGS interior point algorithm for solving strongly convex minimization problems
- Armand, Gilbert, et al.
- 2000
(Show Context)
Citation Context ... , and where f and the components ci are convex and differentiable functions. We solve (1.1) by a quasi-Newton interior point method based on the algorithm introduced by Armand, Gilbert and Jan-Jégou =-=[1]-=- and further developed in [2]. Our new method can start from an infeasible initial point, but it differs from the algorithm proposed in [2] about the feasibility control of the iterates. As a result o... |

13 | Nonlinear equality constraints in feasible sequential quadratic programming
- Lawrence, Tits
- 1994
(Show Context)
Citation Context ...scheme to incorporate equality constraints in methods that solve inequality constraints problems by generating feasible iterates. This scheme has been used in some feasible directions algorithms, see =-=[6, 8]-=- and the recent work of Tits et al. [15]. It consists of keeping the iterates on one side of each equality constraint and penalizing the iterates to force them to go more and more near the boundary of... |

10 |
Feasible directions algorithms for optimization problems with equality and inequality constraints
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- 1976
(Show Context)
Citation Context ...ely driven to 0 thanks to the combination of the logarithmic barrier function and the penalization term. We briefly mention a link between our approach and previous works in nonlinear programming. In =-=[9]-=-, Mayne and Polak proposed a scheme to incorporate equality constraints in methods that solve inequality constraints problems by generating feasible iterates. This scheme has been used in some feasibl... |

6 |
On the existence and convergence of the central path for convex programming and some duality results
- Monteiro, Zhou
- 1998
(Show Context)
Citation Context ...analytic center of the dual optimal set. We call this point the σ -center of the dual optimal set and define it below. Let us first recall the definition of the analytic center of an optimal set (see =-=[11]-=- for related results). Let us denote by opt(P) and by opt(D) the sets of primal and dual solutions of problem (1.1). If opt(P) is reduced to a single point, the analytic center is that20 ARMAND point... |

3 | A BFGS-IP algorithm for solving strongly convex optimization problems with feasibility enforced by an exact penalty approach
- Gilbert, Jan-Jégou
- 2002
(Show Context)
Citation Context ...dition, problem (1.1) has the strict complementarity property, then the whole sequence of outer iterates converges to the analytic center of the primal-dual optimal set. A similar result is proved in =-=[2]-=-, on condition that the sequence of penalty parameters remains bounded. In the present paper, we prove also convergence to a particular point of the primal-dual optimal set. Since the barrier problem ... |