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I have two matrices, $X\in R^{m \times n}, Y \in R^{m \times p}$ and I also have another two matrices, $A \in R^{r \times n}, B \in R^{r \times p}$. Now I want to a matrix $H \in R^{m \times r}$ to satisfy the following objective function. Here H is a common matrix.
The objective function is list as follows:

$f=arg\min_{H}(1-\lambda)||X-HA||_F^2+\lambda||Y-HB||_F^2$, s.t. $H\succeq0$

where X, Y, A, B, and ($0 \leq \lambda \le1$) are known, what is the best way to get H? Can you tell me this kinds of objective function belongs to which problem? matrix factorization? but it is two matrices X and Y. For the constrain of H, how to deal with this? I understand that $\partial f/\partial H = 0$ can deal with this, but I still do not know how to handle H constrain.

Is this H matrix called correlation matrix between X and Y? If X and Y are similarity matrices, can I use the objective function to get H as the correlation matrix to indicate the correlation between X and Y?

Can you help me to get the solution of H?

Any help is appreciated!

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    How can $\rm H$ be positive semidefinite if it is not square? – 2017-02-22
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    @RodrigodeAzevedo --**H** is not required to be positive semidefinite. – 2017-02-22
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    Then why $\rm H \succeq 0$? – 2017-02-22
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    @RodrigodeAzevedo -- $ H \succeq 0 $ means that each entry in H is non-negative. – 2017-02-23

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Given $\lambda \in [0,1]$, we have a convex quadratic program (QP) in $\mathrm X \in \mathbb R^{m \times r}$

$$\begin{array}{ll} \text{minimize} & (1-\lambda) \, \| \mathrm X \mathrm A_1 - \mathrm B_1 \|_{\rm F}^2 + \lambda \, \| \mathrm X \mathrm A_2 - \mathrm B_2 \|_{\rm F}^2\\ \text{subject to} & \mathrm X \geq \mathrm O_{m \times r}\end{array}$$

Vectorizing, we obtain

$$\| \mathrm X \mathrm A_i - \mathrm B_i \|_{\rm F}^2 = \| \mbox{vec} (\mathrm X \mathrm A_i - \mathrm B_i) \|_2^2 = \| \left(\mathrm A_i^{\top} \otimes \mathrm I_m\right) \mbox{vec} (\mathrm X) - \mbox{vec} (\mathrm B_i) \|_2^2$$

Thus, we can write the QP in a more standard form

$$\boxed{\begin{array}{ll} \text{minimize} & (1-\lambda) \, \| \left(\mathrm A_1^{\top} \otimes \mathrm I_m\right) \mbox{vec} (\mathrm X) - \mbox{vec} (\mathrm B_1) \|_2^2 + \lambda \, \| \left(\mathrm A_2^{\top} \otimes \mathrm I_m\right) \mbox{vec} (\mathrm X) - \mbox{vec} (\mathrm B_2) \|_2^2\\ \text{subject to} & \mbox{vec} (\mathrm X) \geq 0_{mr}\end{array}}$$

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    --- Thanks, and first I will learn your method since it is not easy for me (a new person to learn optimization by myself) – 2017-02-23
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    -- can you tell me what existing code can solve this problem you list? Thanks. – 2017-02-23
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    --I prefer MATLAB or R. Thanks. Please show me the code or package if possible. – 2017-02-24
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    Install [CVX](http://cvxr.com). Take a look at the [examples](http://cvxr.com/cvx/examples/). – 2017-02-24