## On a Homogeneous Algorithm for the Monotone Complementarity Problem (1995)

Venue: | Mathematical Programming |

Citations: | 24 - 3 self |

### BibTeX

@TECHREPORT{Andersen95ona,

author = {Erling D. Andersen and Yinyu Ye},

title = {On a Homogeneous Algorithm for the Monotone Complementarity Problem},

institution = {Mathematical Programming},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a generalization of a homogeneous self-dual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "big-M" parameter or two-phase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interior-point and infeasible-starting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and self-dual, infeasible-starting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK-5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...

### Citations

3260 | Convex Analysis
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Citation Context ...0 @ ��f(x=��) \Gammax T f(x=��) 1 A ; (x; ��; s; )s0: A similar augmented transformation was discussed in Ye [33] and it is closely related to the recession function in convex analysis=-= of Rockafellar [26]. Let /(x; ���-=-�) = 0 @ ��f(x=��) \Gammax T f(x=��) 1 A : R n+1 ++ ! R n+1 : (2) Then, it is easy to verify that r/ is positive semi-definite as shown in the following lemma. Lemma 1 . Let rf be positive... |

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Citation Context ...In fact, we do not even know whether such a point exists or not, that is, (MCP ) might be infeasible or feasible but have no positive feasible point. To overcome this difficulty, Ye, Todd, and Mizuno =-=[35]-=- recently developed a homogeneous linear programming (LP) algorithm based on the construction of a homogeneous and self-dual LP model (see Xu et al. [32] for a simplification of the model). More recen... |

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Citation Context ...see Guler and Ye [7]). It is common to choose the step size ff such that the iterates remain in a certain neighborhood or a merit function is decreased to secure global convergence, see Kojima et.al. =-=[11]-=- and Anstreicher and Vial [1]. We have not used this safeguard yet. It is well-known that high-order corrections can be used to improve the practical efficiency of the primaldual barrier algorithm for... |

57 |
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Citation Context ... Related interior-point infeasible-starting algorithms for solving (MCP ) or nonlinear programming problems has been suggested for example by den Hertog et al. [3], El-Bakry et al. [4], Kojima et al. =-=[13]-=-, Kortanek et al. [15], Mizuno et al. [19], Monteiro [20], Monteiro and Wright [23], Tanabe [27], Vial [30], and Wang et al. [31]. 2 A homogeneous MCP model Consider an augmented homogeneous model rel... |

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Citation Context ...overcome this difficulty, Ye, Todd, and Mizuno [35] recently developed a homogeneous linear programming (LP) algorithm based on the construction of a homogeneous and self-dual LP model (see Xu et al. =-=[32]-=- for a simplification of the model). More recently, Ye [34] extended the model to solving the monotone linear complementarity problem, where f is an affine mapping. However, unlike the original LP hom... |

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Citation Context ...pioneered for LP by Fourer and Mehrotra [5] and Turner [28]. The main advantage of this approach is that it has good numerical properties and it is not hampered by a few dense columns in A. Vanderbei =-=[29] propose-=-s an alternative factorization based on the observation that the matrix �� K is quasi-definite. It follows that there exists a factorization of �� K such that D is a diagonal matrix. The main ... |

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Citation Context ...tisfies min(x + j s + j ; �� + + )sfi l (x + ; �� + ) T (s + ;s+ ) n + 1 (41) where fi l is a constant. This prevents the iterates from prematurely to converge to the boundary (e.g., see Guler=-= and Ye [7]-=-). It is common to choose the step size ff such that the iterates remain in a certain neighborhood or a merit function is decreased to secure global convergence, see Kojima et.al. [11] and Anstreicher... |

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Citation Context ... ): From the above two relations, we have (x; ��) T (s 0 ;s0 ) + (s; ) T (x 0 ; �� 0 )s(1 + `)(x 0 ; �� 0 ) T (s 0 ;s0 ): Thus, (x; �� ; s; ) is bounded. The proof of (iv) is well-know=-=n, see McLinden [16, 17] and Kojima,-=- Mizuno, and Noma [12]. We include a version here for completeness. Let (x ; �� ; s ;s) be any maximal complementarity solution for (HMCP ) such that (s ;s) = /(x ; �� ) and (x ) T s + �� ... |

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Citation Context ...long the lines of those proposed by Mehrotra [18] seem to work well. A theoretical result for interior-point algorithms using high-order information was first obtained by Monteiro, Adler, and Resende =-=[22]-=-. Furthermore, Hung and Ye [9] have recently obtained more theoretical results for a high-order version of the homogeneous algorithm for LP. In our preliminary implementation we have used a high-order... |

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Citation Context ...ear programming problems has been suggested for example by den Hertog et al. [3], El-Bakry et al. [4], Kojima et al. [13], Kortanek et al. [15], Mizuno et al. [19], Monteiro [20], Monteiro and Wright =-=[23], Tanabe [27-=-], Vial [30], and Wang et al. [31]. 2 A homogeneous MCP model Consider an augmented homogeneous model related to (MCP ): (HMCP ) minimize x T s + �� subject to 0 @ s 1 A = 0 @ ��f(x=��) \G... |

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Citation Context ...t infeasible-starting algorithms for solving (MCP ) or nonlinear programming problems has been suggested for example by den Hertog et al. [3], El-Bakry et al. [4], Kojima et al. [13], Kortanek et al. =-=[15]-=-, Mizuno et al. [19], Monteiro [20], Monteiro and Wright [23], Tanabe [27], Vial [30], and Wang et al. [31]. 2 A homogeneous MCP model Consider an augmented homogeneous model related to (MCP ): (HMCP ... |

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Interior-point methods for convex programming. Applied Mathematics and Optimization
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Citation Context ... by Zhu [36]. Finally, Potra and Ye [25] extended it for the monotone complementary problem. This condition is included in a more general condition analyzed by Nesterov and Nemirovskii [24] and Jarre =-=[10]-=-. Given x 0 ? 0 and s 0 = f(x 0 ) ? 0 one can develop an interior-point algorithm that generates a maximal complementary solution of the scaled Lipschitz (MCP ) in O( p n log(1=ffl)) interior-point it... |

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Citation Context ... �� K = LDL T (37) using the Bunch-Parlett strategy, where L is a lower triangular matrix and D is a non-singular block diagonal matrix. This approach has been pioneered for LP by Fourer and Mehro=-=tra [5]-=- and Turner [28]. The main advantage of this approach is that it has good numerical properties and it is not hampered by a few dense columns in A. Vanderbei [29] proposes an alternative factorization ... |

21 |
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Citation Context ...ommon to choose the step size ff such that the iterates remain in a certain neighborhood or a merit function is decreased to secure global convergence, see Kojima et.al. [11] and Anstreicher and Vial =-=[1]-=-. We have not used this safeguard yet. It is well-known that high-order corrections can be used to improve the practical efficiency of the primaldual barrier algorithm for LP significantly. Especially... |

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Citation Context ... to a certificate proving infeasibility. Related interior-point infeasible-starting algorithms for solving (MCP ) or nonlinear programming problems has been suggested for example by den Hertog et al. =-=[3]-=-, El-Bakry et al. [4], Kojima et al. [13], Kortanek et al. [15], Mizuno et al. [19], Monteiro [20], Monteiro and Wright [23], Tanabe [27], Vial [30], and Wang et al. [31]. 2 A homogeneous MCP model Co... |

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Citation Context ...al solution (x ; s ) where the number of positive components in (x ; s ) is maximal. Note that the indices of those positive components are invariant among all maximal solutions for (MCP ) (see Guler =-=[6]-=-). Consider a class of (MCP ) where f satisfies the following condition. Let AE : (0; 1) ! (1; 1) be a monotone increasing function such that kX(f(x + d x ) \Gamma f(x) \Gamma rf(x)d x )k 1sAE(ff)d T ... |

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Citation Context ...d by Mehrotra [18] seem to work well. A theoretical result for interior-point algorithms using high-order information was first obtained by Monteiro, Adler, and Resende [22]. Furthermore, Hung and Ye =-=[9]-=- have recently obtained more theoretical results for a high-order version of the homogeneous algorithm for LP. In our preliminary implementation we have used a high-order algorithm similar to Mehrotra... |

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The complementarity problem for maximal monotone multifunction
- McLinden
- 1980
(Show Context)
Citation Context ...orward extension of Kojima et al. [14].) We have (r 0 ; z 0 ) 2 H++ by construction. On the other hand, 0 2 �� H++ from Theorem 2. Thus, ` 0 @ r 0 z 0 1 A 2 H++ : The proof of [ii] is due to McLin=-=den [17], Guler -=-[6] and Kojima et al. [14]. We now prove [iii]. Again, the existence is due to Guler [6] and Kojima et al. [14]. We prove the boundedness. Assume (x; ��; s; ) 2 C(`) then (x; ��) T (r 0 ; z 0 ... |

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A unified approach to infeasible-interior-point algorithms via geometrical linear complementarity problems
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Citation Context ...g algorithms for solving (MCP ) or nonlinear programming problems has been suggested for example by den Hertog et al. [3], El-Bakry et al. [4], Kojima et al. [13], Kortanek et al. [15], Mizuno et al. =-=[19], -=-Monteiro [20], Monteiro and Wright [23], Tanabe [27], Vial [30], and Wang et al. [31]. 2 A homogeneous MCP model Consider an augmented homogeneous model related to (MCP ): (HMCP ) minimize x T s + �... |

17 |
Interior Algorithms for Linear, Quadratic, and Linearly Constrained Non-Linear Programming
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- 1987
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Citation Context ...ed homogeneous model related to (MCP ): (HMCP ) minimize x T s + �� subject to 0 @ s 1 A = 0 @ ��f(x=��) \Gammax T f(x=��) 1 A ; (x; ��; s; )s0: A similar augmented transformation =-=was discussed in Ye [33] and it is close-=-ly related to the recession function in convex analysis of Rockafellar [26]. Let /(x; ��) = 0 @ ��f(x=��) \Gammax T f(x=��) 1 A : R n+1 ++ ! R n+1 : (2) Then, it is easy to verify that... |

13 |
Interior-point methods for nonlinear complementarity problems
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- 1996
(Show Context)
Citation Context ...to be scaled Lipschitz in R n ++ . Such a condition for linearly constrained convex optimization was first proposed by Monteiro and Adler [21] and later generalized by Zhu [36]. Finally, Potra and Ye =-=[25]-=- extended it for the monotone complementary problem. This condition is included in a more general condition analyzed by Nesterov and Nemirovskii [24] and Jarre [10]. Given x 0 ? 0 and s 0 = f(x 0 ) ? ... |

11 |
the Formulation and Theory of the Primal-Dual Newton Interior Point Method for Nonlinear Programming
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- 1992
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Citation Context ...ving infeasibility. Related interior-point infeasible-starting algorithms for solving (MCP ) or nonlinear programming problems has been suggested for example by den Hertog et al. [3], El-Bakry et al. =-=[4]-=-, Kojima et al. [13], Kortanek et al. [15], Mizuno et al. [19], Monteiro [20], Monteiro and Wright [23], Tanabe [27], Vial [30], and Wang et al. [31]. 2 A homogeneous MCP model Consider an augmented h... |

11 |
Computing projections for the Karmarkar algorithm
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- 1991
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Citation Context ...37) using the Bunch-Parlett strategy, where L is a lower triangular matrix and D is a non-singular block diagonal matrix. This approach has been pioneered for LP by Fourer and Mehrotra [5] and Turner =-=[28]-=-. The main advantage of this approach is that it has good numerical properties and it is not hampered by a few dense columns in A. Vanderbei [29] proposes an alternative factorization based on the obs... |

9 |
T.: Limiting behavior of trajectories by a continuation method for monotone complementarity problems
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- 1990
(Show Context)
Citation Context ...ave (x; ��) T (s 0 ;s0 ) + (s; ) T (x 0 ; �� 0 )s(1 + `)(x 0 ; �� 0 ) T (s 0 ;s0 ): Thus, (x; �� ; s; ) is bounded. The proof of (iv) is well-known, see McLinden [16, 17] and Kojima, M=-=izuno, and Noma [12]. We include-=- a version here for completeness. Let (x ; �� ; s ;s) be any maximal complementarity solution for (HMCP ) such that (s ;s) = /(x ; �� ) and (x ) T s + �� = 0; and it is normalized by (r 0 ... |

9 |
A Path-Following Algorithm for a Class of Convex Programing Problems, Zeitschrift fur
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Citation Context ...l 1sff ! 1: Then, f is said to be scaled Lipschitz in R n ++ . Such a condition for linearly constrained convex optimization was first proposed by Monteiro and Adler [21] and later generalized by Zhu =-=[36]-=-. Finally, Potra and Ye [25] extended it for the monotone complementary problem. This condition is included in a more general condition analyzed by Nesterov and Nemirovskii [24] and Jarre [10]. Given ... |

8 |
The Linear Complementarity
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- 1992
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Citation Context ... other words, for every x 1 ; x 2 2 R n + , we have (x 1 \Gamma x 2 ) T (f(x 1 ) \Gamma f(x 2 ))s0: Denote by rf the Jacobian matrix of f , which is positive semi-definite in R n + (see Cottle et al. =-=[2]-=-). (MCP ) is said to be (asymptotically) feasible if and only if there is a bounded sequence f(x t ; s t )g ae R 2n ++ , t = 1; 2; :::, such that lim t!1 s t \Gamma f(x t ) ! 0; where any limit point ... |

8 |
Computational aspects of an interior point algorithm for quadratic programming problems with box constraints
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- 1990
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Citation Context ...uadratic test problems. In addition we use some box-constrained QP problems arising from the so-called obstacle and elasticplastic torsion problems. These problems are taken from the paper Han et al. =-=[8]-=-. The name of these problems all start with the letter B. The B2 problems are slightly modified, because the original problems contain some very large upper bounds on the variables. These bounds are a... |

6 |
On homogeneous and self-dual algorithms for LCP
- Ye
- 1997
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Citation Context ...y developed a homogeneous linear programming (LP) algorithm based on the construction of a homogeneous and self-dual LP model (see Xu et al. [32] for a simplification of the model). More recently, Ye =-=[34]-=- extended the model to solving the monotone linear complementarity problem, where f is an affine mapping. However, unlike the original LP homogeneous model, the model constructed in [34] may not be a ... |

4 |
An Extension of Karmarkar Type Algorithms to a Class of Convex Separable Programming Problems with Global Rate of Convergence
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- 1990
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Citation Context ...: x ? 0g; fl fl X \Gamma1 d x fl fl 1sff ! 1: Then, f is said to be scaled Lipschitz in R n ++ . Such a condition for linearly constrained convex optimization was first proposed by Monteiro and Adler =-=[21]-=- and later generalized by Zhu [36]. Finally, Potra and Ye [25] extended it for the monotone complementary problem. This condition is included in a more general condition analyzed by Nesterov and Nemir... |

3 |
A globally convergent primal-dual interior point algorithm for convex programming
- Monteiro
- 1994
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Citation Context ...r solving (MCP ) or nonlinear programming problems has been suggested for example by den Hertog et al. [3], El-Bakry et al. [4], Kojima et al. [13], Kortanek et al. [15], Mizuno et al. [19], Monteiro =-=[20], Mo-=-nteiro and Wright [23], Tanabe [27], Vial [30], and Wang et al. [31]. 2 A homogeneous MCP model Consider an augmented homogeneous model related to (MCP ): (HMCP ) minimize x T s + �� subject to 0 ... |

3 |
Computational experience with a primal-dual algorithm for smooth convex programming
- Vial
- 1994
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Citation Context ... has been suggested for example by den Hertog et al. [3], El-Bakry et al. [4], Kojima et al. [13], Kortanek et al. [15], Mizuno et al. [19], Monteiro [20], Monteiro and Wright [23], Tanabe [27], Vial =-=[30], and Wang et al-=-. [31]. 2 A homogeneous MCP model Consider an augmented homogeneous model related to (MCP ): (HMCP ) minimize x T s + �� subject to 0 @ s 1 A = 0 @ ��f(x=��) \Gammax T f(x=��) 1 A ; (x... |

3 |
An interior point potential reduction algorithm for constrained equations
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- 1994
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Citation Context ...r example by den Hertog et al. [3], El-Bakry et al. [4], Kojima et al. [13], Kortanek et al. [15], Mizuno et al. [19], Monteiro [20], Monteiro and Wright [23], Tanabe [27], Vial [30], and Wang et al. =-=[31]. 2 A homogeneous MC-=-P model Consider an augmented homogeneous model related to (MCP ): (HMCP ) minimize x T s + �� subject to 0 @ s 1 A = 0 @ ��f(x=��) \Gammax T f(x=��) 1 A ; (x; ��; s; )s0: A simila... |

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A convex property of monotone complementarity problems
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- 1993
(Show Context)
Citation Context ...tion for (HMCP ). Thus, every maximal solution of (HMCP ) must have �� ? 0. Finally, we prove (vi). Consider the set R++ = fs \Gamma f(x) 2 R n : (x; s) ? 0g: As proved by Guler [6] and Kojima et =-=al. [14], R++ is an -=-open convex set. If (MCP ) is strongly infeasible, then we must have 0 62 �� R++ where �� R++ represents the closure of R++ . Thus, there is a hyperplane that separates 0 and �� R++ , that... |