1d
awarded Nice Answer
2d
revised Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$
typo
May
27
comment Why is one of the KKT conditions the same as one of the constraints?
Thanks, @MichaelGrant :D
May
26
comment Why is one of the KKT conditions the same as one of the constraints?
Done. Glad that helped.
May
26
answered Why is one of the KKT conditions the same as one of the constraints?
May
26
comment Constrainted optimization involving logarithms
@NotALoner: if we apply the inequality $\log a + \log b +\log c \le 3\log \frac{a+b+c}{3}$ for the first three terms, then we get a function of $x_4$, which is easy to minimize.
May
26
comment Why is one of the KKT conditions the same as one of the constraints?
(Thus the constraints of the original problem should be added to the KKT conditions because we want to find solutions that satisfy these constraints.)
May
26
comment Why is one of the KKT conditions the same as one of the constraints?
The constraints are part of the KKT conditions, by definition. The idea of KKT conditions (for convex problems) is that: if you solve this system of equations/inequations, independently of the original optimization problem (i.e. forget everything about the original optimization problem: objective function, constraints, etc...), then the obtained solutions are the solutions to the primal and dual of the original optimization problem.
May
26
comment Constrainted optimization involving logarithms
There is a simple solution I think: $\log a + \log b \le 2\log \frac{a+b}{2}$.
May
26
comment Why is one of the KKT conditions the same as one of the constraints?
$\alpha_i$ are the dual variables corresponding to the INEQUALITY constraints, thus non-negative. You might want to learn more about Duality Theory (stanford.edu/~boyd/cvxbook, Chapter 5).
May
22
accepted Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$
May
22
comment Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$
@jf328: Thanks :D
May
21
awarded Curious
May
20
comment Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$
Please have a look at my solution. Thanks.
May
20
answered Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$
May
20
comment Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$
Thanks. I think I have found a solution. I'm going to post it.
May
20
comment Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$
@jf328: Thanks. I was about to post a solution with a similar idea (somehow fixing $x$).
May
20
awarded Tumbleweed
May
20
revised Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$
edited body
May
13
asked Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$
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