Almost Multisecant BFGS Quasi-Newton Method

In convex optimization, quasi-Newton (QN) methods provide an alternative to second-order techniques for solving minimization problems by approximating the curvature of a given target function. This approach reduces computational complexity by relying solely on first-order information while satisfying the secant condition. This paper focuses on multi-secant (MS) extensions of QN, which enhance the Hessian approximation at low cost for both quadratic and non-quadratic problems. Specifically, we use a low-rank perturbation strategy to construct an almost-secant QN method that maintains the positive (semi)definiteness of the Hessian estimate. This property ensures a consistent descent direction and reduces divergence for non-quadratic problems. Our results demonstrate that these perturbations improve the stability and effectiveness of multi-secant updates.