I am new to optimization stuff. I need to formulate and solve this optimization problem.

\(\min \sum_{t\in\mathcal{T}}p_t\)

s.t. \(\sum_{t\in \mathcal{T}}w_t\log_2\left(1+\frac{h}{w_tn_0}p_t\right)= D\)

or

\(\sum_{t\in \mathcal{T}}w_t\ln\left(1+\frac{h}{w_tn_0}p_t\right)= S\)

Here, \(p_t\) is the optimization variable.

Here, \(h\), \(w_t\), \(n_0\) and \(D\)/\(S\) are real and positive and great than \(0\), and they are known. \(\mathcal{T}\) is index set with \(T\) elements, i.e., \(\mathcal{T}={1,2,\cdots, T}\).

Somone please help me to solve this.

How can I express \(p_t\) as a function of others?

asked 13 Oct '17, 02:26

georgefarnandez's gravatar image

georgefarnandez
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accept rate: 0%

edited 13 Oct '17, 11:43

Paul%20Rubin's gravatar image

Paul Rubin ♦♦
14.6k513


I removed the tag "convex-optimization" from your question, because your model is nonconvex as the result of nonlinear equation constraints.

Solving nonconvex problems is generally a pain, but if you only have the one equation constraint, you might look at the Lagrange multiplier method. If you also have bounds on \(p_t\) (for instance, \(p_t\ge 0\)), you should probably look at the KKT conditions.

link

answered 13 Oct '17, 11:48

Paul%20Rubin's gravatar image

Paul Rubin ♦♦
14.6k513
accept rate: 19%

edited 13 Oct '17, 11:49

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Asked: 13 Oct '17, 02:26

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