  Prediction+optimization is a common real-world paradigm where we have to
predict problem parameters before solving the optimization problem. However,
the criteria by which the prediction model is trained are often inconsistent
with the goal of the downstream optimization problem. Recently,
decision-focused prediction approaches, such as SPO+ and direct optimization,
have been proposed to fill this gap. However, they cannot directly handle the
soft constraints with the $max$ operator required in many real-world
objectives. This paper proposes a novel analytically differentiable surrogate
objective framework for real-world linear and semi-definite negative quadratic
programming problems with soft linear and non-negative hard constraints. This
framework gives the theoretical bounds on constraints' multipliers, and derives
the closed-form solution with respect to predictive parameters and thus
gradients for any variable in the problem. We evaluate our method in three
applications extended with soft constraints: synthetic linear programming,
portfolio optimization, and resource provisioning, demonstrating that our
method outperforms traditional two-staged methods and other decision-focused
approaches.
