There is an increasing interest in quantum algorithms for problems of combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best approximation ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. Considering that the Unique Games Conjecture is not valid when there are entangled provers, warm-starting quantum algorithms may allow for an improvement over classical algorithms. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the QAOA is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start.