Solving linear complementarity problems without an LCP solver

----    177 PARAMETER results  

              x           w

i1        0.276
i2        0.256
i3                   28.638
i4        0.152
i5                  108.229
i6                   32.643
i7        0.135
i8        0.536
i9        0.485
i10                  27.609

• MCP models don't have an objective.
• The notation e.x in the model statement means \(e \perp x\).
• The GAMS interface for Knitro does not allow for MCP models.

• QP solvers are more readily available than LCP and MCP solvers.
• A disadvantage of the QP formulation is that we have the matrix \(\color{darkblue}M\) twice in the model.
• It may look like a good idea to introduce the equality \(\color{darkred}w =\color{darkblue}M  \color{darkred}x + \color{darkblue}q\) and use \(\color{darkred}w\) in the model. However this makes the model non-convex: the objective will contain \(\color{darkred}x^T\color{darkred}w\).
• The optimal objective value is zero.
• If \(\color{darkblue}M\) happens to be non-symmetric (this is a bit unusual), we can use the following trick. In the quadratic objective \(\color{darkred}x^T\color{darkblue}M  \color{darkred}x + \color{darkred}x^T\color{darkblue}q\) replace the matrix \(\color{darkblue}M\) by \[\frac{\color{darkblue}M+\color{darkblue}M^T}{2}\] which is symmetric. This will yield the same objective value as the original, non-symmetric version.
• Solving LCPs as QPs is a fairly standard and well-documented approach.

• You may need to add a dummy objective to this problem to be able to feed it to a QCP solver.
• This is convex if \(\color{darkblue}M\) is positive semi-definite.
• We still have two copies of \(\color{darkblue}M\).
• This is SOCP representable [4].

• You may need to add a dummy objective.
• This approach can work even if \(\color{darkblue}M\) is not positive definite and/or not symmetric.
• The main disadvantage of this so-called big-M approach is the need for relatively small upper bounds on \(\color{darkred}x\) and  \(\color{darkred}w\). There are some alternative approaches that can help in this respect.
• We can make the formulation more compact by substituting out \(\color{darkred}w_i\). 
• A first alternative is to use indicator constraints. These are available in many solvers (but not all). This would allow you to write:
  
  

  MIP indicator constraint representation
  \[\begin{align} &\color{darkred}\delta_i=0 \Rightarrow \color{darkred}x_i = 0\\ &\color{darkred}\delta_i=1 \Rightarrow \color{darkred}w_i=0 \\ & \color{darkred}w =\color{darkblue}M  \color{darkred}x + \color{darkblue}q \\ & \color{darkred}x\ge 0,\color{darkred}w\ge 0\\ & \color{darkred}\delta_i \in \{0,1\} \end{align}\]

  
  Unfortunately, GAMS does not really support indicator constraints very well. Specifications for indicator constraints need to be placed in a solver option file. Basically, the GAMS language does not have any syntax for indicator constraints, and the option file approach was really just a hack. (In my opinion, a solver option file should not change the meaning of a model).  In addition, in our example, we need to make make sure the variable \(\color{darkred}\delta\) appears in the model equations. So we need to use another hack to achieve this: just add a constraint \(\sum_i \color{darkred}\delta_i \ge 0\).
• A second alternative is to use SOS1 sets (SOS1 = Special Ordered Sets of Type 1). We can write:
  
  

  MIP SOS1 constraint representation
  \[\begin{align} & (\color{darkred}x_i,\color{darkred}w_i) \in SOS1 \\ & \color{darkred}w =\color{darkblue}M  \color{darkred}x + \color{darkblue}q \\ & \color{darkred}x\ge 0,\color{darkred}w\ge 0\end{align}\]

  
  Having \(\color{darkred}x_i,\color{darkred}w_i\) in a SOS1 set will enforce that only one of them is nonzero. To be precise: we generate \(n=|I|\) sets with 2 members here. Some solvers may generate SOS1 constraints from indicator constraints under the hood.

• There are many ways to solve LCP problems without using a specialized LCP solver.
• The way GAMS handles indicator constraints is a somewhat embarrassing hack. Indicator constraints should be part of the language.
• Some OR papers can benefit from a little bit of extra quality control and some more emphasis on reproducibility.
• There is more to say about this problem, including some critical remarks, than I initially expected.

1. Linear complementarity problem, https://en.wikipedia.org/wiki/Linear_complementarity_problem
2. Richard W. Cottle, Jong-Shi Pang, Richard E. Stone, The Linear Complementarity Problem, Academic Press, 1992.
3. Katta G. Murty. Linear Complementarity, Linear and Nonlinear Programming, http://www-personal.umich.edu/~murty/books/linear_complementarity_webbook/lcp-complete.pdf
4. Second-order cone programming, https://en.wikipedia.org/wiki/Second-order_cone_programming
5. Olvi L. Mangasarian, Linear complementarity as absolute value equation solution, Optimization Letters 8(4), April 2014