Another technique for finding efficient portfolios is called multi-objective (or multi-criteria) optimization. You can use this technique to optimize multiple, often conflicting objectives, such as maximizing returns and minimizing risks, simultaneously. Other examples of multi-objective optimization include:
Aircraft design, requiring simultaneous optimization of weight, payload capacity, airframe stiffness, and fuel efficiency
Public health policies, requiring simultaneous minimization of risks to the population, direct taxpayer costs, and indirect business regulation costs
Electric power generation, requiring simultaneous optimization of operating costs, reliability, and pollution control
Most forms of multi-objective optimization are solved by minimizing or maximizing a weighted combination of the multiple objectives. In the portfolio example, a weighted combination of the return and risk objectives might be:
mean return – (k * standard deviation)
where k > 0 is a risk aversion constant, and the objective is to maximize the function. The relationship between return and risk for the investor is captured entirely by this one function; no additional requirements are necessary.
Geometrically, the optimal solution for a multi-objective function occurs in the saddle point between the optimal endpoints of the individual objectives. In the case of the two-objective function described previously, the optimal solution occurs somewhere on the efficient frontier between the maximum-return portfolio and the minimum-risk portfolio.
For k = 0.5, the optimal solution occurs at the point where the return minus one-half the standard deviation has the highest value.
The following sections describe the model for this problem and its OptQuest solution: