An objective-function-free algorithm for general smooth constrained optimization

A new algorithm for smooth constrained optimization is proposed that never computes the value of the problem’s objective function and that handles both equality and inequality constraints. The algorithm uses an adaptive switching strategy between a normal step aiming at reducing constraint’s infeasibility and a tangential step improving dual optimality, the latter being inspired by the AdaGrad-norm method. Its worst-case iteration complexity is analyzed, showing that the norm of the gradients generated converges to zero like O(1/\sqrt{k+1}) for problems with full-rank Jacobians. Numerical experiments show that the algorithm’s performance is remarkably insensitive to noise in the objective function’s gradient.

💡 Research Summary

The paper introduces ADIC (ADagrad with Inequality Constraints), a first‑order objective‑function‑free (OFFO) algorithm for smooth constrained optimization that never evaluates the objective function value. The method handles both equality and inequality constraints and relies solely on the gradient of the objective, the constraint values, and the Jacobian of the constraints.

The core of ADIC is an adaptive switching mechanism between a “normal” step, which reduces constraint infeasibility, and a “tangential” step, which seeks improvement in the dual sense while staying in the tangent space of the active constraints. Two criticality measures are defined: a dual measure ω_T,k that quantifies the potential benefit of a tangential step, and a primal measure ω_N,k that quantifies the current level of constraint violation. At each iteration the algorithm computes these measures and decides, based on the condition ω_N,k ≤ β·α_T,k·ω_T,k, whether to take a normal step, a tangential step, or both.

The normal step is obtained by solving a trust‑region subproblem on the linearized constraint violation ½‖c(x_k)+J_k s‖², guaranteeing a sufficient decrease of the violation and respecting a bound ‖s_N,k‖ ≤ θ_N ω_N,k. The tangential step is constrained to the null‑space of J_k (J_k s_T,k = 0) and to the non‑negative orthant. Its step size α_T,k follows an AdaGrad‑norm formula α_T,k = η·√(Γ_k + ω_T,k² + σ), where Γ_k accumulates the squares of previous dual measures. Three concrete implementations of the tangential step are proposed: (i) ADIC‑LP, which solves a linear program with an ∞‑norm bound; (ii) a simple scaling of the LP direction; and (iii) ADIC‑P1, which uses the orthogonal projection of the current point onto the tangent space (the “projection‑based” variant). All variants satisfy the descent condition g_T,kᵀ s_T,k ≤ −κ_t α_T,k ω_T,k² and the norm bound ‖s_T,k‖ ≤ θ_T α_T,k ω_T,k.

Convergence analysis is carried out under standard smoothness assumptions (f and c continuously differentiable, bounded below, bounded gradients, bounded constraints) and the full‑rank Jacobian assumption. A Lyapunov function ψ(x,λ)=L(x,λ)+ρ‖c(x)‖ (a sharp augmented Lagrangian) is introduced. Lemmas show that each normal or tangential step yields a sufficient decrease in ψ. The main theorem proves that the average of the combined criticality measure (ω_T,k + ‖c_k‖) decays at the rate O(1/√(k+1)). Consequently, the norm of the generated gradients converges to zero with the same rate, matching the worst‑case complexity of steepest descent or Newton methods for unconstrained problems.

Numerical experiments on a suite of nonlinear problems with both equality and inequality constraints demonstrate the practical performance of ADIC‑LP and ADIC‑P1. The authors compare against existing OFFO methods and classical SQP algorithms. A key observation is that ADIC’s performance is remarkably insensitive to additive Gaussian noise on the gradient (standard deviations ranging from 10⁻³ to 10⁻¹). Because the step size adapts to the accumulated dual measure, the algorithm automatically dampens steps when noise dominates, preserving stability.

In conclusion, ADIC provides a robust, theoretically sound framework for solving smooth constrained problems without ever evaluating the objective function value. By marrying an AdaGrad‑style adaptive stepsize with a trust‑funnel inspired normal step, the method achieves both favorable worst‑case iteration complexity and empirical resilience to gradient noise. Future work may explore extensions to rank‑deficient Jacobians, stochastic gradient estimators, and scalable solution techniques for the linear subproblems in large‑scale sparse settings.