Primal-Dual Interior-Point Methods

• Primal-dual interior-point methods are algorithms for convex optimization that update primal and dual variables along a barrier-defined central path.
• They leverage self-concordant barriers and Riemannian metrics to ensure robust global convergence and efficient short-step iterations.
• Extensions include applications to hyperbolic cone programming, Gaussian quadrature for scaling, and links to quasi-Newton updates.

A primal-dual interior-point method (PDIPM) is a class of algorithms for solving convex optimization problems that simultaneously update both primal and dual variables by following a central path defined by barrier-augmented KKT conditions. These methods extend the original “primal IPM” framework to provide robust path-tracking, efficient global convergence, and analytic complexity guarantees across a broad spectrum of conic, hyperbolic, and nonlinear programming classes. Modern developments in this area have introduced generalizations to hyperbolic cone programming, refined the theory of primal-dual metrics, and connected algorithmic operations to Riemannian geometry, Gaussian quadrature, and quasi-Newton updates (Myklebust et al., 2014).

1. Mathematical Foundations: Primal-Dual Formulation and Self-Concordant Barriers

Primal-dual interior-point methods are defined for convex optimization problems in canonical conic form:

• Primal: $\min\langle c,x\rangle$  s.t. $A x = b$,  $x\in K$
• Dual: $\max\langle b,y\rangle$  s.t. $A^* y+s=c$,  $s\in K^*$

Here $K\subset E$ is a closed, pointed convex cone, $A$ is surjective, and $(x,s)$ are primal/dual variables. A $\theta$-logarithmically homogeneous self-concordant barrier (LHSCB) $A x = b$0 satisfies:

• $A x = b$1
• $A x = b$2

The Hessian $A x = b$3 induces a Riemannian metric $A x = b$4, and the conjugate barrier $A x = b$5 defines an analogous metric on $A x = b$6 (Myklebust et al., 2014).

2. Construction of Local Primal-Dual Metrics and Scaling Operators

A central feature of PDIPMs is the introduction of local “scaling” operators $A x = b$7, mapping $A x = b$8 to $A x = b$9, that facilitate a symmetric treatment of the primal and dual iterates:

• $x\in K$0
• $x\in K$1

Families of admissible scaling operators $x\in K$2 are indexed by tightness of matrix inequalities involving $x\in K$3 and $x\in K$4. For example, the $x\in K$5 family is defined such that $x\in K$6 satisfies

$x\in K$7

with $x\in K$8, $x\in K$9, and $\max\langle b,y\rangle$0 the optimal value of an auxiliary SDP (Myklebust et al., 2014).

3. Short-Step Algorithmic Framework and Iteration Complexity

The short-step PDIPM alternates predictor and corrector steps within a neighborhood of the central path. For a chosen scaling $\max\langle b,y\rangle$1 and centering parameter $\max\langle b,y\rangle$2, the Newton system in scaled variables $\max\langle b,y\rangle$3 is

$\max\langle b,y\rangle$4

This leads (after block-matrix assembly) to a system solved for $\max\langle b,y\rangle$5, followed by a primal-dual update. The step size is chosen so the next iterate remains in the cone interiors.

Key properties:

• Predictor ($\max\langle b,y\rangle$6): reduces gap $\max\langle b,y\rangle$7 by a constant fraction in $\max\langle b,y\rangle$8 iterations, keeping proximity measure small.
• Corrector ($\max\langle b,y\rangle$9, $A^* y+s=c$0): does not reduce $A^* y+s=c$1 but decreases proximity quadratically.

Iteration complexity matches the Nesterov–Todd bound for symmetric cone PDIP: $A^* y+s=c$2 to drive the duality gap below $A^* y+s=c$3 (Myklebust et al., 2014).

4. Extensions: Hyperbolic Cone Programming, Integral Scaling, and Gaussian Quadrature

For hyperbolic cones (e.g., positive semidefinite cones associated with hyperbolic polynomials), the primal barrier $A^* y+s=c$4 admits favorable Hessian estimation properties along segments. Notably, two integral scaling constructs generalize the classical Nesterov–Todd metric:

• Dual integral scaling: $A^* y+s=c$5
• Primal integral scaling: Involves integrating $A^* y+s=c$6, where the integrand is a rational function in $A^* y+s=c$7; this is computed via exact Gaussian quadrature or truncated rules, yielding a computable scaling with provable error guarantees.

Such constructions preserve short-step iteration complexity and exploit cone structure efficiently (Myklebust et al., 2014).

5. Connections to Riemannian Geometry, Operator Means, and Quasi-Newton Theory

The set of admissible primal-dual metrics $A^* y+s=c$8 is geodesically convex in the manifold of positive-definite operators, establishing connections to the Riemannian geometry of self-concordant barriers. The integral scalings can be interpreted as arithmetic means of Hessians (averages over operator spaces), complementary to the Nesterov–Todd geometric mean. Gaussian quadrature, leveraging the polynomial structure of $A^* y+s=c$9, provides computationally efficient and accurate approximations to such operator integrals.

Further, when using midpoint or mean approximations, well-chosen quasi-Newton updates (of DFP or BFGS type) restore the primal-dual metric equations $s\in K^*$0, which aligns interior-point metric updates with classical variable-metric (quasi-Newton) optimization methods. Norm bounds show that such corrections remain controlled within the central-path neighborhood (Myklebust et al., 2014).

6. Algorithmic Impact and Extensions Beyond Symmetric Cones

The developed primal-dual methodology:

• Establishes broad families of short-step PDIPs with explicit local metrics for all self-concordant barriers,
• Extends iteration complexity $s\in K^*$1 beyond symmetric cones to general convex cones,
• Exploits favorable structures of hyperbolic barriers and allows efficient metric computation via operator quadrature,
• Provides geometric and algorithmic ties to other classes of first-order and variable-metric methods.

These advances substantially widen the scope of guaranteed-efficient interior-point technology in convex optimization, set a unified analytic foundation across metric choices, and enable performance gains in conic and hyperbolic programming settings with complicated barrier geometry (Myklebust et al., 2014).