Most of the things in life are constrained optimization problems, whether it is managing a securities portfolio or how fast should you run for how long to cover a specific distance in a specific time. Former involves random variables but the latter is a simple, yet beautiful non linear programming problem. To explore this, I developed a Python application that designs a customized running plan based on user inputs. For example, you can input:

"I want to run 5 km in 25 minutes, alternating between four different paces—10 km/h, 12 km/h, 13 km/h, and 15 km/h. Ensure I don't spend more than 2 minutes running at 15 km/h and I warmup for atleast 7 minutes at 10km/h, and maximize pace variations to keep the run dynamic and engaging, as I prefer frequent speed changes to avoid monotony."

The program uses constrained optimization to compute the ideal time allocation for each pace, ensuring all constraints are met. It's a small yet elegant application of mathematics.

Find the application on: https://run-planner.streamlit.app/

Show me the code: https://github.com/piper-of-dawn/constrained_optimisation_for_running

The program uses constrained optimization to compute the ideal time allocation for each pace, ensuring all constraints are met. It's a small yet elegant application of mathematics.

Mathematical Formulation

1. Decision Variables:
   • Let $x_1, x_2, \ldots, x_n$ represent the time (in hours) spent running at speeds $v_1, v_2, \ldots, v_n$, respectively.
2. Constraints:
   • Distance Constraint: The total distance covered must equal the target distance $D$:
   $$\sum_{i=1}^n v_i \cdot x_i = D$$
   • Time Constraint: The total time spent running must not exceed the available time $T$ (in minutes):
   $$\sum_{i=1}^n x_i = T$$
   • Bounds: Each $x_i$ must lie within a feasible range $[x_{i,\text{min}}, x_{i,\text{max}}]$, representing the minimum and maximum time allocatable to each speed. For example, I want to warmup for 10 mins at 11.5 km/h.
3. Objective Function:
   • Since the primary focus is on satisfying the constraints, the objective function is set as a dummy constant $f(x)=1$. This is a neat trick where you end up solving all the equations in a optimization framework.

Setting Up the Optimization

So we have primary constraint functions:

• Distance Constraint: $$c_1(x) = \sum_{i=1}^n v_i \cdot x_i - D$$
• Time Constraint: $$c_2(x) = \sum_{i=1}^n x_i - \frac{T}{60}$$ and time bounds which we have defined.

Thus we are minimising $f(x) = 1$ subject to above constraints. Remember, the neat trick!

There will likely be multiple solutions to this optimization problem, that is, many pace variations will let you run a distance of $D$ in time $T$. To determine the most suitable running plan, enter a third categorical variable: how frequently you want to vary your pace. Do you prefer to change speeds often, or would you rather run at a steady pace for longer periods? This can be achieved by calculating the Gini coefficient of the time allocations across the different paces in each solution vector. The Gini coefficient is a measure of inequality, and it helps us understand how "even" or "uneven" something is distributed. Here a high Gini coefficient indicates that one or more speeds dominate the running time, which corresponds to fewer pace variations and vice versa. For more details on the Gini coefficient, you can visit the Wikipedia page.