Quantum Algorithms (VQAs) have emerged to obtain possible quantum advantage. In particular, how to effectively incorporate hard constraints in VQAs remains a critical and open question.

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.

Methods for inference and simulation of linearly constrained Gaussian Markov Random Fields (GMRF) are computationally prohibitive when the number of constraintsislarge.

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 study a stochastic bandit problem with a general unknown reward function and a general unknown constraint function. Both functions can be non-linear (even non-convex) and are assumed to lie in a reproducing kernel Hilbert space (RKHS) with a bounded norm.

Methods for query answering over incomplete knowledge graphs retrieve entities that are \emph{likely} to be answers, which is particularly useful when such answers cannot be reached by direct graph traversal due to missing edges. However, existing approaches have focused on queries formalized using first-order-logic. In practice, many real-world queries involve constraints that are inherently vague or context-dependent, such as preferences for attributes or related categories. Addressing this gap, we introduce the problem of query answering with soft constraints. We formalize the problem and introduce two efficient methods designed to adjust query answer scores by incorporating soft constraints without disrupting the original answers to a query. These methods are lightweight, requiring tuning only two parameters or a small neural network trained to capture soft constraints while maintaining the original ranking structure. To evaluate the task, we extend existing QA benchmarks by generating datasets with soft constraints. Our experiments demonstrate that our methods can capture soft constraints while maintaining robust query answering performance and adding very little overhead. With our work, we explore a new and flexible way to interact with graph databases that allows users to specify their preferences by providing examples interactively.

We propose a novel projection method that guarantees the conservation of integral quantities in Physics-Informed Neural Networks (PINNs). While the soft constraint that PINNs use to enforce the structure of partial differential equations (PDEs) enables necessary flexibility during training, it also permits the discovered solution to violate physical laws. To address this, we introduce a projection method that guarantees the conservation of the linear and quadratic integrals, both separately and jointly. We derived the projection formulae by solving constrained non-linear optimization problems and found that our PINN modified with the projection, which we call PINN-Proj, reduced the error in the conservation of these quantities by three to four orders of magnitude compared to the soft constraint and marginally reduced the PDE solution error. We also found evidence that the projection improved convergence through improving the conditioning of the loss landscape. Our method holds promise as a general framework to guarantee the conservation of any integral quantity in a PINN if a tractable solution exists.