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See the definition in http://papers.nips.cc/paper/3646-on-the-convergence-of-the-concave-convex-procedure.pdf. The points generated by an algorithm with this property converge for any initial point....

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It depends. Consider the unconstrained least squares problem $$\min_\mathbf{x}\|\mathbf{a+Bx}\|^2.$$ This problem has the solution $\mathbf{x}^\star = - (\mathbf{B}^H\mathbf{B})^{-1}\mathbf{B}^H\... View answer 1 answers 4 votes 1k views Accepted answer 4 votes As noticed in the comments, your optimization problem can be formulated as $${\rm max}_{\mathbf{x},\mathbf{y}} \mathbf{x}^T S(\mathbf{y}) \mathbf{x} ~~~{\rm s.t.} \|\mathbf{y}\|_{2} = 1,\|\mathbf{x}\|... View answer 3 answers 2 votes 433 views 2 votes The difficulty of your problem is that you try to minimize with respect to a sum of absolute values. If your problem was "just" to minimize with respect to the sum of squared values it would be much ... View answer 2 answers 2 votes 81 views 2 votes Your reasoning is right. However, you have overseen that you can choose y arbitrarily small when x approaches 0. Therefore, the constraints will not be met with equality. For example, you may ... View answer 1 answers 2 votes 63 views Accepted answer 2 votes The answer to your question is yes! You can do this with the help of the Kroecker product, see http://en.wikipedia.org/wiki/Vectorization_(mathematics). Your equality has the form$$ P A R + S A = Q, ... View answer 1 answers 1 votes 101 views 2 votes "I am wondering if for$\lambda$sufficiently large the optimal solution of the second problem approximates arbitrarily close the optimal solution of the first one." In general, the answer is no. ... View answer 2 answers 1 votes 2k views 2 votes If$c$is an integer, it is worth noting that full search is maybe the fastest way to solve your problem. Trying out all possible values$c=1,\dots,4L$is not critical in your case as the problem is ... View answer 1 answers 1 votes 179 views Accepted answer 2 votes there is a solution to your problem: Use the cvx interface for convex programming together with Matlab, see http://cvxr.com/cvx/. You have to download some software and read the user's guide. It's ... View answer 1 answers 1 votes 311 views Accepted answer 2 votes The solution$\mathbf{x}^\star$to your optimization problem is given by$0 < x_j^\star \leq c $for$j = \arg \max_i \frac{a_i}{b_i}$,$x_i^\star = 0$for$i\neq j$. How can we proof this? ... View answer 2 answers 4 votes 2k views 2 votes There is a quick solution to your problem. If you use cvx, you can directly apply the entropy function to formulate your target function$\sum_{i,k}-p_{k,i}*\log{p_{k,i}}$as sum(entr( p )), where p ... View answer 2 answers 4 votes 2k views Accepted answer 1 votes Let us consider equivalence of both problems in the sense that the minimum values are achieved with the same variable vector i.e. both problems have the same minimizer. It is hard to determine the ... View answer 1 answers 2 votes 101 views Accepted answer 1 votes Let me help you with your problem for the case that$|\cdot|$is the Euclidean norm. Regarding$ g(u,v)=g(v)=\inf_{z,x} L(z,v),$we have to minimize$\lVert z\rVert+ v^T (z+b-Ax)$with respect to$...

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It may be hard to find the values for $x,y,z,$ and $\lambda$ analytically. To my best knowledge, this is generally not possible for optimization problems with quadratic cost functions with a quadratic ...

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Your problem has been discussed in the book "Matrix Computations" by Golub and van Loan (http://web.mit.edu/ehliu/Public/sclark/Golub%20G.H.,%20Van%20Loan%20C.F.-%20Matrix%20Computations.pdf). Let ...

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Lets get this done. First, to simplify the probem, let me introduce the following vector notations. I will assume that we have to find $n$ vectors $\mathbf N_i, i = 1,\dots, n.$ We stack all the $\... View answer 1 answers 2 votes 46 views Accepted answer 1 votes Assuming that$c \geq 0$, you can reformulate your problem as a convex optimization problem, introducing a dummy variable$t$. It is equivalent to $$\min_{x \in \Delta_n,t \in \mathbb{R}} \langle g , ... View answer 2 answers 0 votes 200 views 1 votes There is a closed form solution for your problem under the following assumption: x_i > 0, \mu_i > 0 , for all i. You can rewrite \sum_{i=1}^N \log(x_i \mu_i) = \log(\prod_{i=1}^N x_i \mu_i)... View answer 2 answers 5 votes 132 views 1 votes This equation is always true. You proof this by contradiction: 1) It is clear that$$\min_{x\in X,y\in Y}f(x,y)>\min_{x\in X}(\min_{y\in Y}f(x,y))$$cannot hold true. 2) Let us assume$$\min_{x\... View answer 1 answers 0 votes 79 views Accepted answer 1 votes The stated problem is finding the minimum norm point in an underdetermined set of equations, see http://www.math.usm.edu/lambers/mat419/lecture15.pdf This problem is a special case of the least ... View answer 2 answers 1 votes 351 views 1 votes I have devided my answer in two parts. First I introduce the formulation. Second, I consider the optimization. 1) Formulation: For simplicity let me rewrite your problem in vector form and formulate ... View answer 1 answers 0 votes 36 views Accepted answer 1 votes Let me summarize your problem: You want to design$A=\text{diag}(\{a_i\}_{i=1}^n)$and find$\lambda$such that$(AB-I)\lambda = 0$and$|\lambda_i| = 1 \forall i\in \{1,...,n\}$. It is clear that$\...

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There is a solution to your problem: http://inst.eecs.berkeley.edu/~ee127a/book/login/l_ols_variants.html To bring your problem to the form they are using, replace $Dy$ by some vector.

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The difficulty to determine wheather some point $x^\ast$ is optimal depends on the structure of the optimization problem. Non-convex optimization problems: Generally, for non-convex problems, proving ...

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As $A$ is not positive semidefinite and you have (convex) linear equality and inequality constraints, your problem seems to be NP-hard. There is unfortunatly no quick and easy way to solve your ...

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Following Rahuls comment, let us consider the convex optimization problem \begin{equation} \min_{\mathbf{x}} \|\mathbf{Ax}\|_2 ~~~\text{subject to } \mathbf{x} \geq 0, ~~\mathbf 1^T\mathbf x =\...

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An $n \times n$ matrix $Q$ is SPD if $x^T Q x> 0$ for every vector $x$. This is also true the vector $x$ contains zeros. Let now $P= Q_{i:j,i:j}$ some submatrix around the matrix diagonal with $i&... View answer 1 answers 2 votes 55 views 0 votes You have to reformulate the requirement$f(x^\star) = e^{x^\star} -4x^\star = 0$for a root such that the root$x^\star$is a fixpoint of a function$g(x)$, that is$g(x^\star) = x^\star \$. This is ...

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