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I want to solve approximately the following variational problem:

Given a function $c:[-1,1]^2\rightarrow [0,1]$, constants $p_1...p_n\in \mathbb{R}^+$, $\alpha_1...\alpha_n\in \mathbb{R}$, and $\beta_1...\beta_n\in \mathbb{R}$ let

$$V =\sup_{f_i,g_i}\left\{\sum_{i=1}^n \;\;p_i\cdot \int_0^1\int_0^1 f_i(x,y)\cdot g_i(x,y)\; dx\; dy \;\;: \sum_{i}\alpha_i\cdot f_i = c = \sum_i \beta_i\cdot g_i \right\}$$

Where the supremum is taken over all square integrable functions $f_i,g_i:[-1,1]^2\rightarrow [-1,1]$.

Observe that since all functions have domain $[-1,1]^2$ and range $[0,1]$, the value of each integral $\int_{0}^1\int_{0}^1 f(x,y)\cdot g(x,y) \;dx\; dy$ lies between $0$ and $1$. Therefore $V$ lies between $0$ and $\sum_{i} p_i$.

Assumptions:

All functions $f_i$ and $g_i$ and $c$ can be assumed to be square integrable.

If it simplifies the problem, $c:[-1,1]\rightarrow [0,1]$ can be assumed to be $c(x,y)=1$ if $x=y$ and $c(x,y)=0$ otherwise.

Questions:

  1. What numerial methods are available to approximate the value of $V$ up to an $\varepsilon$ additive factor? In other words, I want to find a $V'$ such that $|V-V'|\leq \varepsilon$, where $\varepsilon$ is a given precision parameter.

  2. What is the ratio of convergence of such methods?

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    $\begingroup$ Welcome to SciComp.SE. What are the properties of the functional $V$? What conditions the functions $f, g, c$ satisfy? And, I think that you can rewrite the problem without the need of $p_i$. $\endgroup$ – nicoguaro Nov 25 '15 at 0:36
  • $\begingroup$ thanks. For the moment I would like to impose as few conditions on f and g as possible. The only thing I would like to assume is that f and g are Rieman integrable. $\endgroup$ – verifying Nov 25 '15 at 1:12
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    $\begingroup$ I don't think the explanation why there must be a solution is sound. The set of functions with values in $[0,1]$ is a subset of $L^\infty$, but the functional works on the $L^2$ product of functions. I think that without greater functional analytic care and a statement in which function space the function $c$ is, I'm not convinced that a solution exists. $\endgroup$ – Wolfgang Bangerth Nov 25 '15 at 14:22
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    $\begingroup$ See, the problem with this though is that the set of functions in $L^2([0,1]^2)$ with values in $[0,1]$ is not closed. You can't remove the problem just by declaration. You need to actually prove that there is a solution. $\endgroup$ – Wolfgang Bangerth Nov 25 '15 at 19:41
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    $\begingroup$ A standard approach to try to assert existence of a maximum is to invoke the extreme value theorem, which requires compactness of your feasible set; if a set is compact, as WolfgangBangerth points out, it is also closed. Usually, what is done in the $L^{p}$ context is to invoke a theorem that shows that your set is precompact (or relatively compact, its closure is compact), and then try to show that that it is closed. You need both here; if your set is closed, but not compact, you can't invoke the extreme value theorem to assert existence of an optimum. $\endgroup$ – Geoff Oxberry Nov 25 '15 at 20:35

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