C++ Neural Networks and Fuzzy Logic: Preface

Optimizing a Stock Portfolio

Development of a neural network approach to a stock selection process in securities trading is similar to the

application of neural networks to nonlinear optimization problems. The seminal work of Markowitz in making

a mathematical formulation of an objective function in the context of portfolio selection forms a basis for

such a development. There is risk to be minimized or put a cap on, and there are profits to be maximized.

Investment capital is a limited resource naturally.

C++ Neural Networks and Fuzzy Logic:Preface

Optimizing a Stock Portfolio

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The objective function is formulated in such a way that the optimal portfolio minimizes the objective

function. There would be a term in the objective function involving the product of each pair of stock prices.

The covariance of that pair of prices is also used in the objective function. A product renders the objective

function a quadratic. There would of course be some linear terms as well, and they represent the individual

stock prices with the stock’s average return as coefficient in each such term. You already get the idea that this

optimization problem falls into the category of quadratic programming problems, which result in real number

values for the variables in the optimal solution. Some other terms would also be included in the objective

function to make sure that the constraints of the problem are satisfied.

A practical consideration is that a real number value for the amount of a stock may be unrealistic, as fractional

numbers of stocks may not be purchased. It makes more sense to ask that the variables be taking 0 or 1 only.

The implication then is that either you buy a stock, in which case you include it in the portfolio, or you do not

buy at all. This is what is usually called a zero−one programming problem. You also identify it as a

combinatorial problem.

You already saw a combinatorial optimization problem in the traveling salesperson problem. The constraints

were incorporated into special terms in the objective function, so that the only function to be computed is the