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Convex optimisation problems and tradeoffs in product management

Krishna Kumar K
3 min readSep 3, 2024

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Convex optimisation is a subfield of mathematical optimisation that studies the problem of minimising convex functions over convex sets (or, equivalently, maximising concave functions over convex sets).

A convex optimisation problem is in standard form if it is written as

https://en.wikipedia.org/wiki/Convex_optimization

Note: many product problems are formulated as maximising concave functions

Formulating tradeoffs

Product management involves a lot of tradeoffs. A typical example is trading off user experience and revenue on a search engine. Let’s imagine that there are two product teams — one is tasked with continuously improving user experience with the long term goal of increasing market share and the other is tasked with maximising revenue.

For the first team, they are tasked with maximising user experience. They have learnt that users need relevant results when they search and they do not like irrelevant ads.

They will formulate their goal as—

Maximise (User experience)
Guardrails:
- ad revenue per search ≥ X dollars

The second team will be tasked with maximising ad revenue. However they will definitely have to follow user experience guardrails set by the organisation.

Maximise (ad revenue per search)
Guardrails:
- relevance of ads > Y
- total number of pixels taken up by ads < Z

Formulating goals like this is helpful for communicating tradeoffs. The teams now can work independently with mutually agreed upon guardrails.

This also amenable to reformulation using Lagrangian function which is useful in defining ranking order for search results.

Short term metrics(leading indicators) that lead to long term goals

Another challenge is that most of our important goals are long term. For example, gaining market share through product improvements happens over years. However, we need metrics that can be measured in the short term so that experiment decisions can be made in weeks instead of years.

The practical approach is identifying short term metrics (leading indicators) from behaviours that correlate to long term goals

Source: https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/36299.pdf

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Krishna Kumar K
Krishna Kumar K

Written by Krishna Kumar K

Product Guy. (Worked at Indeed, Microsoft ...). I write about product management, startups, analytics and machine learning. Occasionally I digress...

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