The problem I have can be defined as: $$ \min \frac{1}{2}\mathbf{x}^T\mathbf{Q}\mathbf{x} + \mathbf{c}^T\mathbf{x} $$ s.t. linear equality constraints: $$ \mathbf{Ax=b} $$ and linear inequality constraints: $$ \mathbf{Gx \leq h} $$ $\mathbf{Q}$ is positive semi-definite. The only difference to the regular quadratic programming is that $\mathbf{c}$ is piecewise: $$ c_i= \begin{cases} c^+_i & x_i \geq 0 \\ c^-_i & x_i < 0 \end{cases} $$ so the objective function is continuous at $x_i=0$.

Can someone let me know how to solve this problem ? Is this problem NP hard ? It seems the objective function is not convex anymore and has many possible local minimums. Can we take advantage of fact that all breakpoints are at zeros ? Any help would be greatly appreciated.

The problem I have can be defined as: $$ \min \frac{1}{2}\mathbf{x}^T\mathbf{Q}\mathbf{x} + \mathbf{c}^T\mathbf{x} $$ s.t. linear equality constraints: $$ \mathbf{Ax=b} $$ and linear inequality constraints: $$ \mathbf{Gx \leq h} $$ $\mathbf{Q}$ is positive semi-definite. The only difference to the regular quadratic programming is that $\mathbf{c}$ is piecewise: $$ c_i= \begin{cases} c^+_i & x_i \geq 0 \\ c^-_i & x_i < 0 \end{cases} $$ so the objective function is continuous at $x_i=0$.

Can someone let me know how to solve this problem ? Is this problem NP hard ? It seems the objective function is not convex anymore and has many possible local minimums. Can we take advantage of fact that all breakpoints are at zeros ? Any help would be greatly appreciated.

There's a special property I forgot to mention: $c^+_i > c^-_i$ in my problem.