linearize bilinear quadratic objective terms

Question

I need to model a problem as a linear program. However my working solution contains a (bilinear) quadratic objective term: $$ \sum x_i * y_i \\ x \in \{0,1\} \\ y \in \mathbb{R}^+ $$

The value of $y$ is calculated in a constraint like this: $$ y = \sum\limits_{z \in Z} \sum\limits_{k \in K} f(z,k) $$

Therefore, I tried to move $x$ to the calculation of $y$ and remodel it to an indicator constraint for avoiding quadratic constraints: $$ \begin{align} x_i = 1 &\rightarrow y = \sum\limits_{z \in Z} \sum\limits_{k \in K} f(z,k) \\ x_i = 0 &\rightarrow y = 0 \end{align} $$

This solutions eliminates quadratic objective terms. On the other side is the solution gap around 100% and my originale (quadratic) solution archive <1%.

How can I create a stronger formulation for this case?

Edit: Fixed typo of x in equation like mentioned in the comments.

Update: Based on the answer of Erwin Kalvelagen I linearized the quadratic constraint. The resulting performance of my model is not well and the gap is much higher in the same time like the quadratic constraint.
Since I understand from this questions, it is a bilinear/quadratic constraint. What does this mean for my model exactly? Does this have a significant effect to scalability/performance of models in general? Is it possible to say from experience which solution should be preferred (quadratic or linearized (with bigM-Constraint (?)) version)?

Answer 1

If you know some good bounds, it may be worthwhile to try a formulation with just binary variables. E.g.:

$$\color{darkred}z = \color{darkred}x\cdot \color{darkred}y$$

with $\color{darkred}x \in \{0,1\}$ and $\color{darkred}y \in [0,\color{darkblue}U]$ can be written as:

$$\begin{aligned}& \color{darkred}z \le \color{darkred}x \cdot \color{darkblue}U \\ &\color{darkred}z \le \color{darkred}y\\ & \color{darkred}z \ge \color{darkred}y - \color{darkblue}U \cdot(1-\color{darkred}x)\end{aligned}$$

Sometimes that is faster than indicator constraints (especially if you can find a good, tight $\color{darkblue}U$). No guarantee. The real secret behind developing good integer programming models is just to try many things.