Hello,
Does anyone know of any disadvantage/limitation of using Lagrange multiplier (preferably with reference to papers, thesis, reports, etc.) to get change in objective function value by small amount of change in the constraint?
For example, in the field of water resources optimization, Lagrange multiplier of reservoirs' mass balance constraint can give marginal value of water stored in reservoirs when a net benefit maximization objective function is used. That is how much additional benefit is produced if one additional unit of water is released from the reservoir for downstream demands (e.g. farms).
Is there any flaw in using Lagrangian multiplier to get marginal value of water (from both technical and conceptual points of view)?
Thanks,
Majed
The Lagrange multipliers give you shadow values. (So called because these values cannot be observed directly.) If your model, including your constraints, are well specified, I cannot see why you would suspect a flaw. Having said that, reservoir problems are usually dynamic optimisation problems. For example the value of water changes with the seasons. You store water when it's cheap and release it (for irrigation, say) when it's more valuable. For dynamic optimisation problems you can apply optimal control. I usually recommend to read Dorfman's (1969) classic paper. A recent application to irrigation is Ben-Gal et al. (2012). Dorfman, Robert (1969) An Economic Interpretation of Optimal Control Theory. The American Economic Review 59(5), 817-831.
Ben-Gal, Alon; Weikard, Hans-Peter; Shah, Syed Hamid; van der Zee, Sjoerd E.A.T.M. (2013) Management of irrigation with saline water: accounting for externalities by considering soil-water-plant feedback mechanisms. Water Resources Research 49(5), 2861-2871.
Dear Mahed
As Hans-Peter says Lagrange Multipliers are shadow prices or perhaps better known as Marginal Values
Even when Lagrange Multipliers are shadow prices, the difference is that the second apply to linear functions, that is, they hold along a certain interval.
A Lagrange Multiplier is valid in a point, but probably it will change in the immediate next because it applies to non-linear functions, and therefore there is not an interval where they are constant
Therefore, if your constant is linear you must apply the shadow price, but the Lagrange if it is curve.
This is of course a very elemental explanation, and as Hans-Peter says there are many excellent publications to consult.
Bye the way, thank you Hans-Peter for your explanation related where the 'Shadow' adjective comes from. During many years working with shadow prices I was wondering the meaning of it
I agree with comments of previous comments of Weikard and Munier. Just adding some more discussion on this. Lagrange multipliers are theoretically robust in solving constrained optimization problems. Lagrange multiplier represents a scaling variable that relates the gradient of objective function and constraint surface. That is why it relates to shadow price in economics. It also relates to specific interpretations like constraint forces and relations in physics, This method can be used in differentiable functions that are continuous. I assume this is your question. If the function is discontinuous the calculation with lagrange becomes complex. In addition, if the function is not monotonic or nonconvex, optimization might be difficult as there might be multiple solutions or folds on the functional surface. These are some areas that using Lagrange multipliers will be tricky.
Thank you all for your kind responses. Perhaps I didn't well articulate my question: Is there any case that the Lagrangian multiplier of a reservoir mass balance cannot be used as the water marginal value? If the answer to this question is "Yes" then this can be considered as one of the limitations with Lagrange multipliers.
As far as I understood from your responses, if the objective function is not differentiable and/or discontinuous, then we cannot rely on Lagrangian multiplier as water marginal value. Is that right?
Is there any other condition where Lagrangian multiplier cannot be interpreted as water marginal value?
Thanks
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