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## [Help-glpk] clearing prices in markets with nonconvexities

 From: Robbie Morrison Subject: [Help-glpk] clearing prices in markets with nonconvexities Date: Fri, 14 Jun 2013 06:14:08 +1200 User-agent: SquirrelMail/1.4.22

```Hello all

Some background first.  A common problem in
microeconomics is to determine the clearing in
a commodity market.  If the problem can be
formulated as a linear program, the optimal
solution provides the clearing quantity and the
dual provides the marginal cost and hence the
clearing price.  The method can be extended to
a set of interacting markets under general
equilibrium.

If the underlying structure is mixed-integer (with
the integer variables normally binary) then the
situation is a little more complex.  O'Neill etal
(2002) discuss this solution process in this case:

"Our method for calculating equilibrium prices
is straightforward. First, we solve a MIP to
find the optimal allocation. Next, we remove
the integrality constraints and insert equality
constraints (cuts) that force the integer
variables to assume their optimal values in the
resulting linear program (LP). We then solve
the LP to find the associated dual prices on
the market clearing conditions and added
equality constraints. These dual (or shadow)
prices then can be used as prices to support
a competitive equilibrium. (p271)

I need to implement the addition steps -- namely
the process which adds the cuts and re-solves the
modified problem.

be done on the solver-side.  A new API could be
My first reaction was that this could and should

'glp_cut_and_price'

Otherwise I guess I can do it on the client side
constraints to fix the binary variables at their
current values.  I imagine that this is a
straightforward exercise.

Finally, O'Neill etal (2002) run through the
history of this particular issue of determining
prices under mixed-integer clearing.

REFERENCES

O'Neill, Richard P, Paul M Sotkiewicz, Benjamin
F Hobbs, Michael H Rothkopf, and William R
Stewart Jr.  2002.  Efficient
market-clearing prices in markets with
nonconvexities.  European Journal of
Operational Research.  v164 no1
p269-285.  doi:10.1016/j.ejor.2003.12.011

preprint:
http://www.hks.harvard.edu/hepg/Papers/Oneill.et.al_Efficient.Market.Prices_12-9-02.pdf

---
Robbie Morrison
PhD student -- policy-oriented energy system simulation
Technical University of Berlin (TU-Berlin), Germany