## Interior methods for nonlinear optimization (2002)

Venue: | SIAM Review |

Citations: | 79 - 4 self |

### BibTeX

@ARTICLE{Forsgren02interiormethods,

author = {Anders Forsgren and Philip E. Gill and Margaret H. Wright},

title = {Interior methods for nonlinear optimization},

journal = {SIAM Review},

year = {2002},

volume = {44},

pages = {525--597}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar’s widely publicized announcement in 1984 of a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.

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Citation Context ... subject to semidefiniteness constraints—is arguably the most notable of these problems to receive widespread attention as a direct result of the development of interior methods (see,e.g.,the surveys =-=[65,99,101]-=-). The evident similarity of interior methods to longstanding continuation approaches (see,e.g.,[1,2]) has been recognized since the early days of modern interior methods (see,e.g.,[71]),but numerous ... |

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Citation Context ...define a convergent filter algorithm; we refer the interested reader to [35,36]. Recently,several algorithms were proposed that use a filter to force convergence of a primal-dual interior method (see =-=[8,100,107]-=-). In the primal-dual context,there are several ways to define the filter pair,including the norm of the gradient of the Lagrangian and the proximity norm �C(x)λ − µe�. 6. Treatment of Equality Constr... |

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Citation Context ...n is P T KP = LBL T ,sINTERIOR METHODS 577 where B is now a block-diagonal matrix with diagonal blocks of order 1 × 1or2× 2. Efficient sparse-matrix methods are also available for this case (see,e.g.,=-=[5,29]-=-),but additional overhead is likely since the choice of P depends on numerical considerations unrelated to the sparsity of the factors. In the neighborhood of a trajectory of minimizers we can expect ... |

62 |
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Citation Context |

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Citation Context |

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Citation Context ...rix K of (5.6) is symmetric quasi-definite,which means that for every row and column permutation P there exists a factorization P T KP = LBL T ,where L is unit lower-triangular and B is diagonal (see =-=[53,102]-=-). This allows P to be selected solely for the purposes of obtaining a sparse factor L. For nonconvex problems,the relevant factorization is P T KP = LBL T ,sINTERIOR METHODS 577 where B is now a bloc... |

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Citation Context ...ation (6.6) were applied directly to (6.9). Interior-point algorithms based on the quadratic programming subproblem (6.17) typically utilize SQP-based merit functions to enforce convergence (see,e.g.,=-=[15,27,46,83,123]-=-). If indefiniteness is addressed with a suitable line-search or trust-region technique,the limit points of the sequence of iterates satisfy the second-order necessary optimality conditions for (6.9) ... |

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Citation Context ...3.2) Ilog(x) △ = − m� ln ci(x), also appears to satisfy the desired properties. The earliest history of the logarithmic interior function is worth noting. Frisch’s 1955 “logarithmic potential method” =-=[44]-=- is based on using the gradient of f(x)+ �m i=1 αi ln ci(x) to retain feasibility and accelerate convergence; however,Frisch’s approach did not involve unconstrained minimization of this function. See... |

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Citation Context ...o the optimal face is now completely understood (see,for example,[55]). For convex problems satisfying the LICQ,the barrier trajectory converges to the analytic center of the set of optimal solutions =-=[70]-=-,but the very recent examples of [48] show that the central path can behave in strange and unexpected ways even for infinitely smooth convex functions. Despite these caveats,the “centered” nature of t... |

46 | Superlinear convergence of a stabilized SQP method to a degenerate solution
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Citation Context ...f the Jacobian J E is rank-deficient. Consequently,one needs to modify only H to make the matrix in (6.24) nonsingular [39]. (A related regularization approach for general SQP methods is described in =-=[117]-=-.) We began this section with a discussion of unconstrained minimization of the classical penalty-barrier function Φ PB (6.20),but the techniques presented have all beens586 ANDERS FORSGREN, PHILIP E.... |

44 |
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Citation Context ...trained minimizers of the classical log barrier function converge to a solution of the original constrained problem only if the barrier parameter µ goes to zero. By contrast, modified barrier methods =-=[13, 23,54,76,84]-=- define a sequence of unconstrained problems in which the value of µ remains bounded away from zero,thereby avoiding the need to solve a problem whose Hessian becomes increasingly ill-conditioned as µ... |

40 |
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Citation Context ...n is P T KP = LBL T ,sINTERIOR METHODS 577 where B is now a block-diagonal matrix with diagonal blocks of order 1 × 1or2× 2. Efficient sparse-matrix methods are also available for this case (see,e.g.,=-=[5,29]-=-),but additional overhead is likely since the choice of P depends on numerical considerations unrelated to the sparsity of the factors. In the neighborhood of a trajectory of minimizers we can expect ... |

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(Show Context)
Citation Context ...ation (6.6) were applied directly to (6.9). Interior-point algorithms based on the quadratic programming subproblem (6.17) typically utilize SQP-based merit functions to enforce convergence (see,e.g.,=-=[15,27,46,83,123]-=-). If indefiniteness is addressed with a suitable line-search or trust-region technique,the limit points of the sequence of iterates satisfy the second-order necessary optimality conditions for (6.9) ... |

35 |
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(Show Context)
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(Show Context)
Citation Context ...define a convergent filter algorithm; we refer the interested reader to [35,36]. Recently,several algorithms were proposed that use a filter to force convergence of a primal-dual interior method (see =-=[8,100,107]-=-). In the primal-dual context,there are several ways to define the filter pair,including the norm of the gradient of the Lagrangian and the proximity norm �C(x)λ − µe�. 6. Treatment of Equality Constr... |

34 |
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(Show Context)
Citation Context ...trictly feasible,but nonlinear, paths emanating from the origin. An important consequence of the MFCQ is that its satisfaction at a first-order KKT point implies boundedness of the set of multipliers =-=[45]-=-. Lemma 2.11 (implication 1 of MFCQ: a bounded multiplier set). If ¯x is a firstorder KKT point at which the MFCQ is satisfied, then the set of multipliers Mλ defined in (2.5) is bounded. Proof. First... |

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(Show Context)
Citation Context ...x case is thus how to treat indefiniteness,a topic that would consume too many pages to allow its detailed treatment in this article. A more fundamental difficulty arises at the theoretical level. In =-=[106]-=-,a class of one-dimensional examples was recently defined for which certain line-search barrierSQP methods (section 6.2.2) fail to find a feasible point and,much more seriously,do not even converge to... |

30 |
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(Show Context)
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30 |
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(Show Context)
Citation Context ...of augmented Lagrangian methods for nonlinearly constrained optimization. We consider only shifts of the form (6.26b); for more general discussions of shifted constraints in interior methods,see,e.g.,=-=[23,46,60,84,98]-=- and section 4.6. The appeal of altering the constraints via either slacks or shifts may not be evident at first,since the dimension of the problem appears to have increased. However,two major benefit... |

30 |
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(Show Context)
Citation Context ...ion of a linear program (2.10); verifying assumption (iii) for all p such that J ∗ A p ≥ 0 requires finding the global minimizer of a possibly indefinite quadratic form over a cone,an NP-hard problem =-=[75,82]-=-,not to mention the issue of how to check that (iii) holds for all λ ∈Mλ. If,however,the gradients of the active constraints at x ∗ are linearly independent and strict complementarity holds,Theorem 2.... |

29 |
Solving symmetric indefinite systems in an interior-point method for linear programming
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(Show Context)
Citation Context ...HC (except when the constraints are simple bounds,in which case HC is simply H plus a positive diagonal matrix). An alternative strategy is to factorize the full (n+m)×(n+m) system in (5.2) (see,e.g.,=-=[43,49,50]-=-),typically after symmetrizing the system. A symmetric matrix can be created by multiplying the second block of equations in (5.2) by Λ−1 and changing the sign of the second block of columns,giving (5... |

29 | On practical conditions for the existence and uniqueness of solutions to the general equality quadratic progarmming problem - Gould - 1985 |

29 |
Ill-conditioning and computational error in interior methods for nonlinear programming
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(Show Context)
Citation Context ... has been repeatedly observed that the computed solutions in interior methods are almost always much more accurate than they should be (see,for example, [90,110]). To explain why,numerous papers,e.g.,=-=[41,85,113,114,116,118,119]-=-, have examined ill-conditioning in several contexts,including linear programming,linear complementarity,and primal-dual systems for nonconvex nonlinear optimization. One of the surprising results tha... |

26 | On the global convergence of trust-region SQP-Filter algorithms for general nonlinear programming
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(Show Context)
Citation Context ...d to the filter,and all filter entries that it dominates are removed. In practice many details need to be specified in order to define a convergent filter algorithm; we refer the interested reader to =-=[35,36]-=-. Recently,several algorithms were proposed that use a filter to force convergence of a primal-dual interior method (see [8,100,107]). In the primal-dual context,there are several ways to define the f... |