Ordinary Lipschitz-Hessian analysis uses a fixed Euclidean scale and is not well adapted to logarithmic barriers, whose curvature diverges near the boundary. Self-concordance controls the third derivative relative to the local Hessian metric and makes Newton's method quantitatively invariant to this changing scale.
Self-concordance in the local Hessian norm
Definition. A \(C^3\) convex function \(F\) on an open convex domain is self-concordant if \[ |D^3F(x)[h,h,h]| \le2\bigl(D^2F(x)[h,h]\bigr)^{3/2} \qquad(x,h). \] When \(H_x=\nabla^2F(x)\succ0\), define \[ \|h\|_x=(h^\top H_xh)^{1/2},\qquad \|u\|_{x,*}=(u^\top H_x^{-1}u)^{1/2}. \]
Hessian variation. If \(F\) is self-concordant and \(\|h\|_x<1\), then \[ (1-\|h\|_x)^2H_x \preceq H_{x+h} \preceq(1-\|h\|_x)^{-2}H_x. \]
Proof. For a fixed direction \(v\), let \(\phi(t)=v^\top H_{x+th}v\). Polarization of the self-concordance inequality yields \[ |\phi'(t)|\le2\|h\|_{x+th}\phi(t). \] Applying the defining inequality first to \(h^\top H_{x+th}h\) shows \(\|h\|_{x+th}\le\|h\|_x/(1-t\|h\|_x)\). Integrating the differential inequality for \(\log\phi(t)\) from \(0\) to \(1\) gives the two matrix bounds.
Newton decrement and damped steps
Definition. The Newton direction and decrement are \[ p_N(x)=-H_x^{-1}\nabla F(x),\qquad \lambda(x)=\|p_N(x)\|_x =\|\nabla F(x)\|_{x,*}. \]
Self-concordant Newton estimates. Let \(F\) be closed, strictly convex, and self-concordant, with minimizer \(x^\star\). If \(\lambda(x)<1\), then \[ F(x)-F(x^\star) \le-\lambda(x)-\log(1-\lambda(x)). \] The damped step \[ x^+=x+\frac1{1+\lambda(x)}p_N(x) \] satisfies \[ F(x)-F(x^+) \ge\lambda(x)-\log(1+\lambda(x)). \] Once \(\lambda(x)\) is sufficiently small, the full Newton step is feasible and its next decrement is of order \(\lambda(x)^2\).
Proof outline. Restrict \(F\) to the Newton ray and integrate the Hessian-variation bounds. They produce the one-dimensional inequalities \[ F(x+h)\le F(x)+\nabla F(x)^\top h+ [-\|h\|_x-\log(1-\|h\|_x)] \] for \(\|h\|_x<1\), together with the corresponding lower bound. Substitute \(h=tp_N\) and minimize the upper bound in \(t\in[0,1/\lambda)\), obtaining \(t=(1+\lambda)^{-1}\) and the damped decrease. Comparing with the unknown minimizer gives the gap bound. Applying Hessian variation after a full step controls the new gradient and yields the quadratic decrement recursion.
Logarithmic barriers and the central path
For a convex problem \[ \min_x f(x)\quad\text{subject to}\quad g_i(x)\le0, \] the logarithmic barrier is \[ \Phi(x)=-\sum_{i=1}^m\log(-g_i(x)) \] on the strict interior. For \(t>0\), the central point \(x(t)\) minimizes \[ tf(x)+\Phi(x). \] Its first-order equation is \[ t\nabla f(x(t)) +\sum_{i=1}^m\frac{-1}{g_i(x(t))}\nabla g_i(x(t))=0. \] Defining \(\lambda_i(t)=-1/(t g_i(x(t)))>0\) turns this into dual stationarity, with \[ \lambda_i(t)(-g_i(x(t)))=\frac1t. \] Thus the aggregate complementarity gap is \(m/t\).
Barrier suboptimality. For a convex problem satisfying Slater's condition, a central point satisfies \[ 0\le f(x(t))-f^\star\le\frac mt. \]
Proof. The multipliers \(\lambda_i(t)\) are dual feasible and stationarity makes \(x(t)\) minimize the Lagrangian. Weak duality gives \[ f^\star\ge \inf_x\mathcal L(x,\lambda(t)) =\mathcal L(x(t),\lambda(t)) =f(x(t))+\sum_i\lambda_i(t)g_i(x(t)) =f(x(t))-\frac mt. \] Primal feasibility gives the other inequality.
Polynomial complexity of path following
A \(\nu\)-self-concordant barrier permits a short-step path-following method to increase the central-path parameter by a factor \(1+O(\nu^{-1/2})\) while restoring centrality with a bounded number of Newton steps. Reaching duality gap \(\varepsilon\) therefore requires \[ O\!\left(\sqrt\nu\log\frac{\nu}{\varepsilon}\right) \] Newton iterations. For the standard logarithmic barrier of \(m\) scalar inequalities, \(\nu\) is of order \(m\). The Hessian metric becomes singular at the boundary, so the local unit ball remains inside the feasible domain and guides iterates through its interior.