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Business Math: The Problem of the Local Maxima

29 Jun

Hill climbing is a mathematical optimization discipline based on local search principles that proposes a model for finding solutions to a problem by iteratively changing elements of it. In that sense, when attacking a problem using hill climbing model, we will initially select an arbitrary solution and subsequently will try to improve it by, iteratively, changing a single element of it.

A classic example of hill climbing is represented by the traveling salesman problem on which Given a list of cities and their pairwise distances, the task is to find the shortest possible route that visits each city exactly once and returns to the origin city. From a mathematical perspective (for us math geeks) hill climbing attempts to optimize a target function   , where   is a vector of continuous and/or discrete values. At each iteration, hill climbing will adjust a single element in   and determine whether the change improves the value of  .

No, you are not reading the wrong weblog J It turns out that hill climbing is one of the mathematical models that has a direct applicability in business and problem solving. However, hill climbing models can also influence business with one of their main deficiencies: the problem of the local maxima.

By always optimizing a single element of the solution in order to find a better solution, hill climbing models guarantee to find a solution to a problem but can’t guarantee to find the best solution. As a business analogy, when trying to solve a problem with no clear solution path, we can select a random small solution that addresses part of the problem; let’s say we climb a himm. If the solution makes things better, find a random solution to the next part of the problem (we climb another hill) and keep climbing and climbing. If it doesn’t, try another random solution to the same part of the problem and repeat the same steps. Using this model, will are always guarantee to go higher and higher by focusing on one problem at a time. The main deficiency of this model, is that prevents you from focusing on finding the optimal solution. What happens if you are just climbing a hill that is in front of the mountain when your goal is to climb the mountain?

Even without knowing its foundation, hill climbing is an intuitive model to solve problems in startups. Consequently, new companies are constantly victims of the problem of the local maxima. Even though there is no magic solution to this problem there are a few things you can do to mitigate its effects. As a founder CEO, the best way to fight the problem of the local maxima is to lay out a clear vision for the specific product or service you are providing and constantly re-evaluate strategies with your teams in order to accomplish that vision. This technique will help to influence (this is the best you can hope for) your team to not always select a random solution to the problem but sometimes evaluate that short term solution (climbing a hill) towards the bigger vision (getting to the top of the mountain).

Going back to math land, there are variations of the simple hill climbing model that we can apply when implementing this strategy. For instance, the “steepest hill climbing model” selects a solution to the immediate problem (climbing the hill) by evaluating its proximity to the final solution. Another variance to consider is “scholastic hill climbing” on which, after selecting a random solution to the local problem, improvement is evaluated against a certain criteria. If the evaluation is not successful, the algorithm goes back and selects another random solution to the local problem.

Whether you are a math geek or not, hill climbing is one of the intuitive problem solving techniques we all use in startups. As a founder CEO, it’s important to understand that, sometimes, you are going to be the victim of the problem of the local maxima. However, as long as your team keeps the overall vision of the product or company present in every decision they make, you will have a good shot at climbing that mountain.

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Posted by on June 29, 2012 in Uncategorized

 

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