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I have a function with two variables say $$g(x,y)=f(x)−h(x,y)\ $$ where $$f(x)= ax-bx^2\ $$ and $$h(x,y)=(x+y)^2\ $$ and $$ y\ge0, x+y\ge0.$$ My purpose is to maximize $g(x,y)$ for $x$, simultaneously minimizing $h(x,y)$ for $y$. How I can do this? Is there any literature available related to this problem? Waiting your expert response.

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2 Answers 2

For a treatment of problems of this kind, see

Simultaneous Maximisation in Economic Theory, 2013,

This document provides translations of two important but neglected articles of Bruno de Finetti, Problemi di "optimum" and Problemi di "optimum" vincolato. The articles deal with the theory of simultaneously maximising a number of functions. A major field of application is economic theory, which through the assumption of rational, maximising behaviour of all participants defines a set of interdependent maximum problems as its model of the world.

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Notice that generally the problem is ill-posed: it is generally not true that the maxima of one function will correspond to minima of another, so "simulatenous maximize and minimize" doesn't make sense.

What you can do is to solve one of the optimization problems, and hope that one of the optimal points of $h$ is also one of $g$. In this case this strategy will work: obviously $h(x,y)$ has global minima along the line $x=-y$. We are left with $$\max_x g(x,-x) = ax-bx^2$$ which, after taking the derivative and setting it equal to zero, gives $$a - 2bx = 0.$$

So the point you want is $$\left(\frac{a}{2b}, -\frac{a}{2b}\right).$$

EDIT: The inequalities don't change much. The minimizers of $h$ are still the points on the line $y=-x$ for $y\geq 0$ (do you see why?) Then optimizing $g$ amounts to $$\max_x g(x, -x) = \max_y g(-y, y) = -ay - by^2$$ subject to $y \geq 0$. Can you solve this on your own?

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I apologize, I forget to mention one thing. I edit the question – Lucky Sep 24 '13 at 15:54
I edited the question, Can u please reconsider it and give me solution – Lucky Sep 24 '13 at 15:56
Thanks, I will solve. Is there any book/paper that can help me to study such type of problems in details? – Lucky Sep 24 '13 at 16:01
There are many good books on optimization; I would start by studying convex optimization. Here's a textbook available online: – user7530 Sep 24 '13 at 16:04
This type of problem is almost always ill-posed (does not make sense) so I don't know if much is written about it. The strategy to use is as I outlined above: solve one optimization problem while completely ignoring the other one, and then check the global optima of that problem and hope one of them is a global optimum of the other. – user7530 Sep 24 '13 at 16:05

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