# Optimization Problem with a Penalty Factor

I am working on an investment optimization problem where I'm trying to maximize returns over a 20-year period with a given total budget. The investment involves an initial capital and annual contributions. While the main goal is to maximize the net profit, I also want to disincentivize investing an overly high initial capital in order to preserve liquidity.

Here's the basic setup:

• I have a total budget of $$B=€50000$$ to be allocated over $$T=20$$ years.
• The annual net return rate is fixed at $$R=0,026 \equiv 2,6\%$$ (compounded annually).
• The decision variables are the initial capital ($$P$$) and the annual contribution ($$C$$).

I'm considering introducing a penalty for the initial capital using a penalty factor $$\lambda$$ in the objective function.

My initial thought is to use:

$$max(A-\lambda P)$$

where $$A$$ represents the total accumulated amount after 20 years exploiting compound interest, and it is given by the following formula:

$$A=P(1+R)^{T}+C[\frac{(1+R)^{T} - 1}{R}]$$

The constraints of my optimization problem are:

$$P+T\cdot C=B =50000 \\ P \leq B=50000\\ C \leq \frac{B}{T} = \frac{50000}{20} = 2500\\ P\geq0\\ C\geq0$$

My questions are:

1. Is this an effective way to introduce such a penalty?
2. How can I determine the best value for $$\lambda$$ to make this disincentive meaningful without overshadowing the primary objective of maximizing returns?
3. Are there any other common or more effective methods to introduce this kind of disincentive in optimization problems?

You have two separate objectives: maximizing terminal value $$A$$ and minimizing (or at least reducing) initial investment $$P.$$ You might want to search the web using the phrases "bicriterion optimization" or "multiobjective optimization" to find a variety of approaches people use, none of which are definitively the "best" or "most correct" choice.
Using a penalty term is definitely one option. As for choosing $$\lambda,$$ there is no "best value". You can try solving the model over a range of values of $$\lambda$$ and see what tradeoff most appeals to the decision maker (you or whoever has the final say).
Another possibility is to use lexicographic ordering of the objectives, which is a fancy way of saying that you first solve the model to get the maximal $$A$$ with no penalty on $$P,$$ then solve a modified version of the model in which you constrain $$A$$ to be at least the optimal value (which of course means equal to it) and minimize $$P,$$ looking for the smallest initial investment that yields the optimal return.
Yet another possibility is to maximize $$A$$ (again without penalty on $$P$$), find out what the optimal value of $$P$$ is, then add an upper bound less than that on $$P$$ and solve again. Repeat this for a variety of progressively lower bounds on $$P,$$ then plot the results to get a sense of the relationship between $$P$$ and $$A$$ and let the decision maker choose their preferred tradeoff.