  Benchmarks in the utility function have various interpretations, including
performance guarantees and risk constraints in fund contracts and reference
levels in cumulative prospect theory. In most literature, benchmarks are a
deterministic constant or a fraction of the underlying wealth; as such, the
utility is still a univariate function of the wealth. In this paper, we propose
a framework of multivariate utility optimization with general benchmark
variables, which include stochastic reference levels as typical examples. The
utility is state-dependent and the objective is no longer
distribution-invariant. We provide the optimal solution(s) and fully
investigate the issues of well-posedness, feasibility, finiteness and
attainability. The discussion does not require many classic conditions and
assumptions, e.g., the Lagrange multiplier always exists. Moreover, several
surprising phenomena and technical difficulties may appear: (i) non-uniqueness
of the optimal solutions, (ii) various reasons for non-existence of the
Lagrangian multiplier and corresponding results on the optimal solution, (iii)
measurability issues of the concavification of a multivariate utility and the
selection of the optimal solutions, and (iv) existence of an optimal solution
not decreasing with respect to the pricing kernel. These issues are thoroughly
addressed, rigorously proved, completely summarized and insightfully
visualized. As an application, the framework is adopted to model and solve a
constraint utility optimization problem with state-dependent performance and
risk benchmarks.
