Team Decision Problems with Convex Quadratic Constraints

TL;DR

This paper establishes that linear decision strategies are optimal for a broad class of team decision problems with quadratic objectives and constraints, applicable in stochastic and deterministic contexts, and provides methods to compute these strategies.

Contribution

It proves the optimality of linear decisions in quadratic team problems with multiple constraints and develops semidefinite programming approaches for their computation.

Findings

Linear decisions are optimal for infinite and finite team problems with quadratic constraints.

Optimal strategies can be computed via semidefinite programming.

The theory applies to dynamic linear quadratic team decision problems.

Abstract

In this paper, we consider linear quadratic team problems with an arbitrary number of quadratic constraints in both stochastic and deterministic settings. The team consists of players with different measurements about the state of nature. The objective of the team is to minimize a quadratic cost subject to additional finite number of quadratic constraints. We first consider the problem of countably infinite number of players in the team for a bounded state of nature with a Gaussian distribution and show that linear decisions are optimal. Then, we consider the problem of team decision problems with additional convex quadratic constraints and show that linear decisions are optimal for both the finite and infinite number of players in the team. For the finite player case, the optimal linear decisions can be found by solving a semidefinite program. Finally, we consider the problem of minimizing a quadratic objective for the worst case scenario, subject to an arbitrary number of deterministic quadratic constraints. We show that linear decisions are optimal and can be found by solving a semidefinite program. Finally, we apply the developed theory on dynamic team decision problems in linear quadratic settings.