Continuous OptimizationFeasibility problems with complementarity constraints☆
Introduction
A Mathematical Programming Problem with Complementarity Constraints (MPCC) (Luo, Pang, Ralph, 1996, Outrata, Kocvara, Zowe, 1998, Ralph, 2007) can be defined in the form where y ∈ m, while and are continuously differentiable functions. The feasible set of MPCC will be denoted by D and min {x, w} denotes a vector of components min {xi, wi}, . For all the variables xi, wi are said to be complementary and satisfy:
MPCC has appeared frequently in optimization models and has significant applications in different areas of science, engineering and economics (Luo, Pang, Ralph, 1996, Outrata, Kocvara, Zowe, 1998, Ralph, 2007). Many theoretical and application papers in Operations Research, as well as survey papers on related topics (Bomze, 2012, Chen, 2000, Júdice, 2014, Kovacevic, Pflug., 2014, Lin, Fukushima, 2010), have been devoted to this problem in recent years. For example, transport network models were considered in García-Rodenas and Verastegui-Rayo (2008), Walpen, Mancinelli and Lotito (2015), Wu, Yin and Lawphongpanich (2011), bilevel optimization in Kovacevic and Pflug (2014), variational inequality formulations in Toyasaki, Daniele and Wakolbinger (2014), multiobjective problems with complementarity constraints in Lin, Zhang and Liang (2013), Ye (2011), electricity markets in Ehrenmann and Neuhoff (2009), Guo, Lin, Zhang and Zhu (2015), Hu and Ralph (2007), Yao, Oren and Adler (2007), quadratic programming with complementarity constraints in Ralph and Stein (2011), optimality conditions in Pang (2007), order-value applications in Andreani, Dunder and Martínez (2005), and oligopolistic equilibrium in Yao, Adler and Oren (2008), among others.
Clearly, MPCC can be seen as a Nonlinear Programming Problem where the n complementarity constraints are replaced with (2) or even with x ⩾ 0, w ≥ 0. Attempts for solving MPCC by means of nonlinear programming algorithms present some difficulties, mainly because these algorithms may converge to points from which there exist obvious first-order descent directions. This issue is a consequence of the so-called double zeros or biactive indices, i.e., feasible points satisfying at least a constraint with both variables xi and wi equal to zero. These difficulties have motivated much research on weak forms of stationarity (Ferris, Pang, 1997, Hoheisel, Kanzow, Schwartz, 2013, Luo, Pang, Ralph, 1996, Outrata, Kocvara, Zowe, 1998, Ralph, 2007, Scheel, Scholtes, 2000 Outrata, Kocvara, Zowe, 1998, Ralph, 2007, Scheel, Scholtes, 2000) and several algorithms have been designed to compute such weak stationary points (Anitescu, 2005, Toyasaki, Daniele, Wakolbinger, 2014, Anitescu, Tseng, Wright, 2007, Benson, Sen, Shanno, Vanderbei, 2006, Fang, Leyffer, Munson, 2012, Fletcher, Leyffer, 2004, Fukushima, Luo, Pang, 1998, Fukushima, Tseng, 2002, Hoheisel, Kanzow, Schwartz, 2013, Hu, Ralph, 2004, Jiang, Ralph, 2003, Júdice, Sherali, Ribeiro, Faustino, 2007, Leyffer, López-Calva, Nocedal, 2006, Luo, Pang, Ralph, 1996, Outrata, Kocvara, Zowe, 1998, Ralph, 2007).
In this paper, we will discuss how to compute a feasible solution of the MPCC, that is, a solution of the following Horizontal (possibly nonlinear) Complementarity Problem (HNCP) Gowda (1995):
We will assume that so that the number of equations in (3) is smaller than or equal to the number of unknowns. The case in which has been studied in Andreani, Júdice, Martínez and Patrício (2011b). The case of H affine has been thoroughly discussed in the literature (see for instance Júdice (2014) for a recent survey). The HNCP is NP-hard in this case Murty (1988) but there are many MPCCs where finding a single feasible solution can be considered as an easy task Júdice (2014).
The problem of finding a feasible point of MPCC at which the objective function achieves a target value ct is naturally formulated as follows:
Note that the problem (4) can be written as a standard HNCP adding two auxiliary variables v1 and v2, as follows:
In this paper we will extend the algorithm introduced in Andreani et al. (2011b), which deals with the case for the underdetermined HNCP (3) where r may be smaller than . The Projected-Gradient Underdetermined Newton-like algorithm (PGUN) combines fast interior-point iterations with projected-gradient steps. A line-search procedure is employed guaranteeing sufficiently reduction of the natural merit function Andreani, Júdice, Martínez and Patrício (2011a) associated to HNCP. This will allow us to establish global convergence of the PGUN algorithm to a solution of HNCP or to a stationary point of the merit function with a positive function value. In this case the algorithm terminates unsuccessfully. Fast local convergence will be established under reasonable hypotheses.
Computational experience with PGUN for solving the HNCP associated to feasible solutions of some MPCC test problems from a well-known collection Leyffer (2000) will show that, for many instances, projected-gradient iterations are seldom used and the algorithm is able to converge very fast to a solution of HNCP. For other instances, PGUN converges slowly using projected-gradient iterations to a stationary point of the merit function that seems not to be a solution of the HNCP. A practical criterion will be introduced to stop prematurely PGUN and avoid many projected-gradient iterations. As the natural merit function is nonconvex, the choice of the starting point is very important for the success of PGUN. Here we will suggest to restart the PGUN algorithm with a new initial point whenever the criterion mentioned before forced the algorithm to stop prematurely. Numerical results with an implementation of PGUN incorporating these two practical procedures (premature stopping criterion and restarting) show that the method is in general efficient to solve the HNCP and seems to perform better than a Projected Levenberg-Marquardt algorithm Kanzow, Yamashita and Fukushima (2005). We have also tested PGUN for solving (5) associated to a target ct equal to the best known objective function value of some MPCCs from the collection mentioned before. As discussed in Fernandes, Friedlander, Guedes and Júdice (2001), the introduction of the target constraint to HNCP makes this problem more difficult to tackle and PGUN has more difficulties to solve the HNCP in this case. Despite this, PGUN has been able to provide a target feasible solution of MPCC for the large majority of tested instances.
The organization of this paper is as follows. The properties of the merit function for the HNCP are studied in Section 2. The algorithm PGUN will be described and its global convergence will be analyzed in Section 3. Section 4 will be devoted to the local convergence of the PGUN algorithm. Computational experience with the PGUN algorithm will be reported in Section 5 and some conclusions will be presented in the last section of the paper.
Notation: The 2-norm of vectors and matrices will be denoted by ‖·‖. If there is no risk of confusion we denote as it has been already done in Section 1. We adopt the usual convention of denoting X the diagonal matrix whose entries are the elements of x ∈ n. The Moore–Penrose pseudoinverse of the matrix A will be denoted by A†. The Jacobian matrix of Φ: n → m, with components will be defined by
We define and
The Interior of this set will be denoted by Int(Ω).
Section snippets
Stationary points of the sum of squares
The HNCP (3) may be expressed in the form where is given by and has continuous first derivatives.
We define the natural merit function: and we consider the problem where Ω is defined in (6). From now on we will denote .
It is well known that, if z* is an unconstrained stationary point of “Minimize ‖Φ(z)‖2” and the residual Φ(z*) is not null, then the rows of
Projected gradient underdetermined Newton-like algorithm and global convergence
In this section we introduce a Projected Gradient Underdetermined Newton-like (PGUN) Algorithm for the solution of the (possibly) underdetermined system (8). This algorithm is an extension of the method introduced in Andreani et al. (2011b) for the solution of this system when the number of equalities is equal to the number of variables, i.e., when . PGUN generates iterates lying inside Int(Ω) and combines interior-point Newton-like and projected-gradient directions with a line-search
Local convergence
At Step 2 of PGUN one considers the linear system given by (17) and (18). If this linear system is incompatible the algorithm goes to Step 4 where a projected gradient direction is computed. All along this section we will assume that, whenever (17)–(18) is compatible, the computed direction dk will be the minimum-norm solution of that system. This implies that dk belongs to the range space of F′(zk)⊤ and where μk ≥ 0 satisfies (19).
Note that the minimum-norm
Computational experience
In this section we will report some experiments with the PGUN algorithm for the solution of (3) and (5). In order to have a better idea of the efficiency of PGUN in practice, we have compared the PGUN method with the Projected-Gradient Levenberg–Marquardt (PLM) algorithm Kanzow et al. (2005).
Conclusions
In this paper, we introduced a Projected-Gradient Underdetermined Newton-like (PGUN) algorithm for computing a feasible solution of a Mathematical Programming Problem with Complementarity Constraints (MPCC). The algorithm can also be applied for the computation of a feasible solution of MPCC that satisfies a certain objective function target. In both cases the algorithm searches a solution of an associated Horizontal Complementarity Problem (HNCP). It was shown that PGUN is globally convergent
Acknowledgment
We are indebted to the associate editor and two anonymous referees for helpful remarks that improved a lot the quality of this paper.
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This work was supported by PRONEX-Optimization (PRONEX - CNPq / FAPERJ E-26 / 171.164/2003 - APQ1), FAPESP (Grants 06/53768-0, 2012/10444-0 and CEPID Industrial Mathematics 2011/51305-0 ) and CNPq.
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The research of Joaquim J. Júdice was partially supported in the scope of R&D Unit UID/EEA/50008/2013, financed by the applicable financial framework FCT/MEC through national funds and when applicable co-funded by FEDER - PT2020 partnership agreement.