Chapter 2 Literature Review
2.2 MPCC Solution Algorithms
Mathematical programs with complementarity constraints (MPCC) is a subclass of the mathematical programs with equilibrium constraints (MPEC) the first appearance of which dates back to the 1960s in the work of Kirch¨gassner[76]. Since then, the MPCC has be an subject of intense investigation. Detailed studies of MPCCs and related problems can be found in the books by Luo et al.[50] and Outrata et al.[77].
Scheel and Scholtes[78] review several stationarity conditions of MPCCs, which are the foundation of MPCC solution algorithms. One important stationarity concept, the strong-stationarity is stated in Section 3.2.1. Slightly different from the frequently addressed complementarity constraints between two functions, their stationarity condi- tions are presented for the more general case of vertical complementarity constraints, where more than two component functions are involved in each complementarity constraint.
In order to solve an MPCC,
MPCC: min
x f(x) (2.1)
s. t. g(x) ≤ 0
0 ≤ F(x) ⊥ G(x) ≥ 0,
one may reformulate it into a nonlinear program. Intuitively, this can be achieved through converting the set of complementarity constraints 0 ≤ F(x) ⊥ G(x) ≥ 0 (as shown in Figure 2.1 (a)) into the following sets of inequality constraints:
F(x) ≥ 0, G(x)(x) ≥ 0, F(x) ◦ G(x) ≤ 0 (2.2)
where the symbol ◦ represents the Hadamard product, i.e., the term-by-term product operation between two vectors: a ◦ b = [a1, · · · , an]T◦ [b1, · · · , bn]T = [a1b1, · · · , anbn]T. In the last few decades several local algorithms have been
proposed for finding a stationary point for an MPCC problem. Since the reformulated problem is a nonlinear program, the first numerical approaches for locally solving MPCCs employed traditional nonlinear programming (NLP) algo- rithms. However, as discussed in Chen and Florian[79], an important sufficient condition for the stability of a nonlinear program, the Mangasarian-Fromovitz constraint qualification (MFCQ)[11], is violated for all feasible points of this problem. The failure of MFCQ may have important negative numerical implications: the multiplier set is unbounded, the active constraint normals are linearly dependent, and a linear relaxation of the reformulated nonlinear program problem can become inconsistent, arbitrarily close to a solution to the optimization problem[80]. Therefore, some
i ] [F 0 ] [ ] [ 0≤Fi⊥Gi≥ (a) (b) i ] [G [G]i i ] [F 0 , ] [ ] [ 0 ] [ , ] [ → ≤ ≥ k k i i i i t t G F G F
Figure 2.1: Feasible Space of 0 ≤ [F]i⊥ [F]i≥ 0 and Its Regularization.
difficulties may arise when an MPCC is solved through NLP algorithms.
Significant efforts have been made to investigate MPCC solution algorithms over the past decade, mostly belonging to four categories: regularization methods, complementarity penalty methods, interior point methods, and sequential quadratic programming (SQP).
Regularization and Complementarity Penalty Methods
The regularization presented by Scholtes[81] convert the set of complementarity constraints 0 ≤ F(x) ⊥ G(x) ≥ 0 into the following sets of inequality constraints:
F(x) ≥ 0, G(x)(x) ≥ 0, F(x) ◦ G(x) ≤ tke (2.3)
where tk> 0 is a regularization parameter, and e = (1, 1, · · · , 1)T ∈ Rp.
The idea behind the regularization approach is to solve a sequence of regularized problems corresponding to a sequence {t}k→ 0, to obtain a sequence of local optimizers {x}k(as shown in Figure 2.1 (b)). Scholtes[81] presents
several necessary conditions for the limit point of {x}k to be stationary for the original MPCC. His computational
results indicate that regularization can be more effective than direct application of a solver to the naive nonlinear pro- gram formulation (Equation (2.2)). Ralph and Wright[82] further investigate the properties of Scholtes regularization formulation together with several other regularization formulations.
The regularization method is closely related to the complementarity penalty method[83], in which the violation of the complementarity constraints is penalized in the objective function:
Ppen(γk) : min x f(x) + γkF(x) TG(x) (2.4) s. t. g(x) ≤ 0 F(x) ≥ 0, G(x) ≥ 0,
whereγk> 0 is a penalty parameter.
Similar to the regularization approach, the penalized formulation is solved under a sequence of penalty parameters {γ}k→ ∞, and the conditions under which the limit point of the obtained sequence of local optimizers is stationary
for the original MPCC are established by Hu and Ralph[83]. It is also shown that the penalized problem can be mirrored to the regularized problem given γk= tk−1. Additionally, Huang et al.[84] reformulate the complementarity
constraints via the nonsmooth Fischer-Burmeister function and employs a smooth penalty functions to treat the derived nonsmooth constrained program.
Interior Point Methods
Interior point methods can be applied to some of the above mentioned reformulations of the original MPCC problem. Leyffer et al.[85] present an interior point method based on the penalized problem in Equation (2.4):
Ppen(γk, µk) : min x f(x) + γkF(x) TG(x) − µ k mg
∑
i=1 ln(si) − µk p∑
i=1 [ln(Fi(x)) + ln(Gi(x))] (2.5) s. t. gi(x) + si= 0where s is a vector of nonnegative slack variables corresponding to the inequality constraints.
The resulting parameterized problem is solved under a sequence of γk, and µkwith {γ}k→ ∞ and {µ}k→ 0. Leyffer
et al.[85] present the global and local convergence results for the interior point algorithm, while Ralph[86] points out that the results only hold when the constraint functions are linear. Interior point methods based on other reformulations of the original problem are also available: Luo et al.[50] discuss interior point methods for MPCCs; Liu and Sun[87] and Ragunathan and Biegler[88] provide interior methods under weaker assumptions. Shanbhag[89] proposes an interior point method based on the reformulation in Equation (2.2) and shows that it converges to a second-order stationary point. DeMiguel et al.[90] discuss a two-sided relaxation scheme and provide local convergence theory for an interior method coupled with such a relaxation scheme.
Sequential Quadratic Programming
As mentioned previously, it has been known for years that the reformulation in Equation (2.2) fails to satisfy MFCQ, and failures have been reported apply existing SQP solvers to it. Therefore, it was quite a surprise when Fletcher and Leyffer[91] applied a filtered SQP to a suite of test problems and finds that the method usually converged superlinearly to a local solution. This is the first time superlinear convergence was observed for a standard NLP method applied directly to an equivalent nonlinear program reformulation of an MPCC problem. Fletcher and Leyffer[80] explain the superlinear convergence result of the SQP applied to the equivalent reformulation; and Anitescu[92] provides the
global convergence theory for SQP methods.
Other Approaches
Giallombardo and Ralph[93] propose a piecewise decomposition methods that applies an NLP algorithm at each iterate to a feasible piece of the MPCC, which contains the current incumbent solution. Some global convergence results are presented assuming that the embedded NLP solver implements a trust region scheme. In addition, Jiang and Ralph[94] propose a smoothing approach for MPCC, by smoothing the complementarity constraints with a differentiable version of the Fischer-Burmeister function, as given by
ψ([F]i, [g]i) =
q
[F]2i + [G]2i + τ − [F]i− [G]i (2.6)
The smoothed problem is solved either with sequential SQP solutions with a reducing τ sequence or with an interior point approach. Smoothing approaches with other expression have also been proposed[95].
In addition to the above contributions, some commercial software packages for MPCC solution are also available. KNITRO[96], developed by Ziena Optimization Inc., is a collection of nonlinear optimization methods that support complementarity constraints. The NLPEC solver of GAMS[97] automatically reformulates MPCC problems into nonlinear programs, and calls NLP solvers to solve the resulting nonlinear programs.