Transport model as a variational inequality emp JAMS GAMS short %MODELSTAT.OPTIMAL%] 'LP not solved'; *------------------------------------------------------------------------------- * That's how the VI looks like positive variable dPrice(j) 'demand price'; negative variable sPrice(i) 'supply price*(-1)'; equations ggrad(i,j) 'MCP version of grad from VI'; ggrad(i,j).. c(i,j) - sPrice(i) - dPrice(j) =N= 0; model mcpTransport / ggrad.x, supply.sPrice, demand.dPrice /; $ontext *Adopt solution from LP solve and verify it is a solution of the MCP sPrice.l(i) = supply.m(i); dPrice.l(j) = demand.m(j); mcpTransport.iterlim = 0; solve mcpTransport using mcp; abort$[mcpTransport.objVal > 1e-6] 'Input for model mcpTransport should be optimal, was not'; $offtext *------------------------------------------------------------------------------- * Now use EMP to this reformulation * F(x) = c for our VI: LP models yield a linear VI equations grad(i,j) 'dcost/dx(i,j)'; grad(i,j).. c(i,j) =N= 0; model viTransport / grad, supply, demand /; file myinfo / '%emp.info%' /; put myinfo '* complementarity pairs for grad.x' /; putclose 'vi grad x'; $onecho > jams.opt subsolveropt 1 $offecho $onecho > path.opt crash_method none; crash_perturb no; lemke_start first; $offecho viTransport.optfile = 1; solve viTransport using emp; abort$[viTransport.objVal > 1e-6] 'Input for model viTransport should be optimal, was not'; ]]> Example showing how to write a VI as an MCP We want to write the VI using EMP to avoid manual translation to MCP. We use the definitions of VI and MCP from Steven P. Dirkse, Ph.D. Dissertation Robust Solution of Mixed Complementarity Problems. Mathematical Programming Technical Report 94-12, August 1994. ftp://ftp.cs.wisc.edu/math-prog/tech-reports/94-12.ps Pages 4-6 In this case, the VI to start with is what we get by letting F(x) = df/dx, where f is the objective in the transport model. We adjusted the data to get nonzero supply marginals. Contributor: Steven Dirkse and Jan-H. Jagla, January 2009