find coefficients that maximize an objective function

Question

I have an object function, $Obj(t)$, that I want to minimize. I have $1000$ test cases that returns different outcomes for $t$. The test cases can return different outcome even with the same $t$ value. Each of $1000$ test cases have input variables $x_i$, and the $t$ value for each test case is decided by $\sum \alpha_i x_i$. Here, I want to decide $\alpha_i$ that minimize my object function $Obj(t)$.

Using a real world example, I want to sell a home very quickly while maximizing the selling price. Obviously, the lower price will make the selling process quicker but it will lower the selling price. I have a record of 1000 homes selling history which records the hypothetical selling prices and corresponding time to take to sell home (not really practical though). Each of 1000 homes has various features, such as size, years built, .... From the features, I want to extract the best selling price for my objective function, and I need to find coefficients of each feature to maximally satisfy my objective function. Any possible mathematical approach for the problem?

Answer 1

You could treat selling price as an additional feature (independent variable), and let selling time be the target (dependent variable). Run a linear regression to get the coefficients $\alpha_i$. Substitute the known feature values $x_i$ into the resulting multilinear function to obtain a linear relationship between selling price and selling time. Now you can decide the tradeoff between price and time. For example, you could use the selling price that corresponds to the largest acceptable selling time.