Analytic Center Cutting Plane Methods

• ACCPM is an iterative convex optimization method that refines a polyhedral approximation using analytic centers to solve complex feasibility and variational inequality problems.
• It integrates separation oracles and logarithmic barrier optimization to dynamically generate cutting planes, enhancing convergence and solution precision.
• The method is applied in copositive programming, complete positivity detection, and wireless resource allocation, demonstrating efficient empirical performance.

The Analytic Center Cutting Plane Method (ACCPM) is an iterative convex optimization paradigm that leverages interior-point principles and cutting-plane techniques to solve feasibility, optimization, and variational inequality problems over convex sets, particularly in settings characterized by uncertain or complex polyhedral constraints. ACCPM algorithms operate by maintaining and iteratively refining a polyhedral approximation to the feasible set, guided by the analytic center of this polyhedron, and by incorporating information via separation oracles that produce new cutting planes. This approach has found direct application in copositive programming, matrix complete positivity, variational inequality problems, and learning-based resource allocation in network systems.

1. Mathematical Framework and Formulation

ACCPM constructs an outer approximation of a convex feasible region using a sequence of polyhedral sets or intersections with convex cones, parameterized by linear inequalities (cuts). At each iteration, the method computes the analytic center of the current localization polyhedron—defined as the minimizer of a self-concordant logarithmic barrier function corresponding to the active cut constraints. This is formalized for a polyhedron

$$K_k = \{ x \in \mathbb{R}^n : a_i^T x \leq b_i,\, i=1,\dots,k \},$$

with analytic center

$$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$

The method is naturally extended to settings where the feasible set is the intersection of a ball and a convex cone, such as

$$\min \langle C, X \rangle \quad \text{subject to } \| \mathrm{vec}(X) \|^2 \leq r^2,\, X \in \mathrm{COP}^d,$$

for copositive programming (Badenbroek et al., 2020).

2. Barrier Function Optimization and Analytic Center Computation

The ACCPM analytic center is obtained by minimizing the logarithmic barrier function over the current feasible polyhedron or intersection:

$$\Phi(x) = -\log(r^2 - \| x \|^2) - \sum_{i=1}^m \log(b_i - a_i^T x),$$

with gradient and Hessian given by

$$\nabla \Phi(x) = \frac{2x}{d} + \sum_{i=1}^m \frac{a_i}{s_i}, \quad \nabla^2 \Phi(x) = \frac{2}{d}I + \frac{4}{d^2}xx^T + \sum_{i=1}^m \frac{a_i a_i^T}{s_i^2},$$

where $d = r^2 - \|x\|^2$, $s_i = b_i - a_i^T x$. The analytic center is typically approximated via a damped Newton method, with step size determined to ensure that the iterates remain strictly feasible with respect to all active slacks (Badenbroek et al., 2020, Zeng, 2017).

3. Separation Oracles and Cut Generation

At each central iteration, an external oracle is invoked to either (i) certify the current analytic center as feasible—whereupon a supporting cut can be generated to cut off suboptimal solutions—or (ii) provide a violated constraint (deep cut) to further refine the feasible approximation. In copositive or complete positivity settings, this oracle reduces to solving a mixed-integer linear program (MILP) characterizing copositivity:

$\min \{ y^T X y : \mathbf{e}^T y = 1, y \geq 0 \},$

achievable via precise MILP formulations (Badenbroek et al., 2020). In continuous optimization under parametric uncertainty (e.g., interference channel learning (Tsakmalis et al., 2015)), observed feedback is translated into a pair of inequalities, which are then added to the polyhedral localization set.

4. Algorithmic Implementation and Iterative Procedure

The generic ACCPM workflow is as follows:

1. Initialization: Initialize the feasible localization (e.g., polyhedron, ball, or intersection).
2. Compute analytic center: Use Newton-type inner iterations to determine current analytic center.
3. Oracle call: Query the separation oracle with the center; either certify or return a violated cut.
4. Add cut and update: Refine the feasible set by including the new cut.
5. Convergence check: Evaluate optimality or feasibility gap; terminate or repeat.

Pseudocode in the copositive matrix setting (Badenbroek et al., 2020) and variational inequality context (Zeng, 2017) follows a similar skeleton, with problem-specific adaptations for the separation oracle and stopping criteria.

5. Convergence Theory and Complexity

Classical ACCPM theory (cf. Goffin–Luo–Ye, 1996) ensures that, with sufficiently deep cuts and a self-concordant barrier, the number of analytic center iterations to reach an $\varepsilon$-approximate solution in $n$ variables is $$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$0, where the hidden factor absorbs moderate polylogarithmic terms. For optimization over a bounded localization set containing a ball of radius $$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$1 and inscribed in a ball of radius $$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$2, the analytic center algorithm requires no more than $$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$3 iterations ($$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$4 barrier parameter) (Zeng, 2017).

Empirical results for complete positivity detection observe $$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$5 scaling in the number of expensive oracle calls as a function of matrix dimension $$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$6, outperforming ellipsoid-based schemes by an order of magnitude for moderate sizes ($$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$7) (Badenbroek et al., 2020). In high-dimensional resource allocation (e.g., $$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$8 users in power control), convergence to 1% solution error is achieved in $$x^{(k)} = \arg\min_{x \in K_k} h(x), \quad h(x) = -\sum_{i=1}^k \ln(b_i - a_i^T x).$$9 analytic center updates (Tsakmalis et al., 2015).

6. Practical Applications

ACCPM has demonstrated versatility across several advanced problem domains:

• Completely Positive Matrix Detection: Characterizing membership in the CP cone via optimization over the copositive cone, with direct implications for nonconvex quadratic programming and matrix factorization (Badenbroek et al., 2020).
• Variational Inequality Problems: Solving (pseudo/quasi)monotone VIs over convex bodies, with extension to unbounded settings via nested polyhedral localizations (Zeng, 2017).
• Learning-Based Wireless Resource Allocation: Joint learning and constraint satisfaction for interference-limited secondary network power control, where each probing action yields a refinement of the feasible model for unknown interference characteristics (Tsakmalis et al., 2015).

The method naturally incorporates domain-specific separation oracles—including MILP copositivity checkers and data-driven binary/quantized feedback interpreters—and handles polyhedral and conic constraints in unified fashion.

7. Implementation Considerations and Empirical Performance

Robust implementations combine efficient Newton solvers for analytic center computation, specialized MILP engines for non-polyhedral oracles, and pruning based on Dikin ellipsoids to maintain relevant cut sets and control computational growth (Badenbroek et al., 2020). ACCPM demonstrates favorable empirical scaling and convergence behavior compared to alternative schemes such as the ellipsoid method and center-of-gravity cutting plane method—often requiring significantly fewer central iterations for a given solution precision.

The approach is extensible to any convex program for which a separation oracle (or approximate cut generator) can be specified, provided that a containing ball for the feasible region is known or can be bounded.

8. Extensions and Generalizations

The ACCPM framework accommodates a broad suite of extensions, including:

• General Copositive Programming: Directly applicable to general linear objectives over the intersection of the copositive cone, ball constraints, and additional linear inequalities, with suitable oracle generalization (Badenbroek et al., 2020).
• Unbounded Domains: Via a sequence of increasingly large polytopes to outer-approximate unbounded feasible sets, with proof of finite termination or subsequence convergence under strong monotonicity conditions (Zeng, 2017).
• Data-Driven and Learning Settings: Adapted to situations where constraints are dynamically learned or inferred (e.g., wireless communications), with ongoing cut enrichment driven by real-time feedback (Tsakmalis et al., 2015).

In summary, the Analytic Center Cutting Plane Method supports a rigorous and algorithmically efficient approach to high-dimensional convex optimization and feasibility problems with complex or partially unknown constraint structure, offering both theoretical and practical advantages across diverse applications (Badenbroek et al., 2020, Zeng, 2017, Tsakmalis et al., 2015).