Populating a user interface using quadratic constraints

A computer-implemented method comprising:

• determining a decision space representing a set of content items to be presented on a user interface of a social networking site, the decision space accounting for competing quadratic constraints and interaction effects;
• estimating the decision space to linearize the competing quadratic constraints;
• determining, in the estimated decision space and using an objective function, a display probability for each content item in the set of content items, each respective display probability corresponding to a given content item's probability of display in a specific content slot of a plurality of content slots on the user interface;
• and
• causing display of the content items with the highest display probabilities.

A non-transitory computer-readable medium storing executable instructions thereon, which, when executed by a processor, cause the processor to perform operations including:

• determining a decision space representing a set of content items to be presented on a user interface of a social networking site, the decision space accounting for competing quadratic constraints and interaction effects;
• estimating the decision space to linearize the competing quadratic constraints;
• determining, in the estimated decision space and using an objective function, a display probability for each content item in the set of content items, each respective display probability corresponding to a given content item's probability of display in a specific content slot of a plurality of content slots on the user interface;
• and
• causing display of the content items with the highest display probabilities.