[Paper Review] DC3: A learning method for optimization with hard constraints

Priya L. Donti, David Rolnick+1 more|arXiv (Cornell University)|Apr 25, 2021

Constraint Satisfaction and Optimization29 references64 citations

TL;DR

DC3 learns to solve optimization problems with hard equality and inequality constraints by completing partial solutions and applying differentiable corrections, achieving feasible, near-optimal results faster than traditional solvers, including in AC optimal power flow tasks.

ABSTRACT

Large optimization problems with hard constraints arise in many settings, yet classical solvers are often prohibitively slow, motivating the use of deep networks as cheap "approximate solvers." Unfortunately, naive deep learning approaches typically cannot enforce the hard constraints of such problems, leading to infeasible solutions. In this work, we present Deep Constraint Completion and Correction (DC3), an algorithm to address this challenge. Specifically, this method enforces feasibility via a differentiable procedure, which implicitly completes partial solutions to satisfy equality constraints and unrolls gradient-based corrections to satisfy inequality constraints. We demonstrate the effectiveness of DC3 in both synthetic optimization tasks and the real-world setting of AC optimal power flow, where hard constraints encode the physics of the electrical grid. In both cases, DC3 achieves near-optimal objective values while preserving feasibility.

Motivation & Objective

• Motivate fast approximations to large optimization problems with hard constraints where classical solvers are slow.
• Develop a neural-network-based framework that guarantees feasibility with respect to equality constraints via equality completion.
• Introduce a differentiable inequality-correction step to enforce inequality constraints while maintaining equality feasibility.
• Demonstrate DC3 on convex QPs, a simple non-convex problem, and AC optimal power flow (ACOPF).
• Show that DC3 yields feasible solutions with near-optimal objective values and substantial speedups over baselines.

Proposed method

• Output a partial set of decision variables that can be completed to satisfy equality constraints via a function ϕx; backpropagate through ϕx using implicit differentiation when explicit gradients are unavailable.
• Apply a differentiable gradient-based correction ρx to reduce inequality violations while remaining on the equality-feasible manifold.
• Train the network with a soft loss that includes objective value and constraint violations: ℓsoft(ŷ)=fx(ŷ)+λg||ReLU(gx(ŷ))||22+λh||hx(ŷ)||22.
• During testing, complete the partial outputs to a full vector that satisfies equalities, then apply correction to satisfy inequalities.
• Maintain differentiability of both completion and correction to enable end-to-end backpropagation through the constraint enforcement steps.

Experimental results

Research questions

• RQ1 Can a neural network learn to predict a partial solution that can be completed to satisfy equality constraints?
• RQ2 Can a differentiable correction procedure ensure inequality feasibility while preserving equality feasibility?
• RQ3 How does DC3 perform in convex, simple non-convex, and real-world non-convex (ACOPF) settings in terms of optimality and feasibility?
• RQ4 How does DC3 compare to traditional optimizers and other deep-learning-based methods in terms of speed and feasibility?

Key findings

• DC3 preserves feasibility for both equality and inequality constraints across convex QPs, simple non-convex tasks, and ACOPF, while achieving near-optimal objective values.
• DC3 is significantly faster than differentiable QP solvers (about 78x faster than qpth) and about 9x slower than optimized OSQP in convex QP settings; in non-convex tasks, DC3 is ~9–10x faster than IPOPT with competitive objective values.
• In ACOPF, DC3 yields feasible, near-optimal results and runs about 10x faster than the PYPOWER optimizer.
• Ablations demonstrate that removing either equality-completion or inequality-correction leads to constraint violations, underscoring the necessity of both components.