# Solution of non-convex smooth function with non-convex constraints

I am trying to solve the following minimization problem \begin{equation*} \begin{aligned} & \underset{x,y}{\text{minimize}} & & H(x,y) \\ & \text{subject to} & & f(x,y) \leq \lambda \end{aligned} \end{equation*}

where $H = ||t -\sum_k x_{k}f(y_{k}) - \sum_k x_{k}y_{k}||_2^2$ is smooth, non-convex function and $f = |\sum_k x_{k}f(y_{k})|$ is non-smooth, non-convex function, but it is convex if we keep one variable constant. I was thinking of using Proximal Alternating Linearized Minimization, is this a valid approach? Is there any other approach to solve the above problem? A related problem is Proximal operator of multiplication of two variables

• Are $x,y$ reals? Or vectors? – Michael Jun 8 '18 at 4:29
• @Michael $x$ and $y$ are vectors – Dushyant Sahoo Jun 8 '18 at 4:34
• Too general. Give us $H$ and $f$. – Rodrigo de Azevedo Jun 8 '18 at 13:17
• A special case of your problem is the unconstrained minimization of a non-convex function $H(x)$ for a vector $x \in \mathbb{R}^n$, and there is no efficient solution for that general problem. If $x,y$ were scalars and $H$ is Lipschitz continuous, I would suggest discretizing the space $[0,1]\times [0,1]$ into squares of size $\epsilon^2$ and testing all $1/\epsilon^2$ grid points. – Michael Jun 8 '18 at 14:50
• @RodrigodeAzevedo I wanted to know if there exists a general algorithm for solving such class of problems. Edited the question. – Dushyant Sahoo Jun 8 '18 at 16:10