Outer Approximation Algorithms

• Outer approximation algorithms are iterative methods that construct and refine convex relaxations to enclose the feasible set or Pareto front.
• They employ supporting hyperplanes, valid cuts, and duality principles to systematically narrow down the solution space in convex, MINLP, and global optimization.
• Modern variants incorporate warm-start strategies, quadratic cuts, and distributed protocols to enhance performance in large-scale optimization problems.

Outer approximation algorithms are a foundational class of iterative methods in convex, mixed-integer, and multiobjective optimization—central to both theory and practice across vector optimization, MINLP, SDP, PDE-constrained control, global optimization, and decentralized estimation. The unifying principle is the iterative construction of outer (superset) polyhedral or convex relaxations which enclose the desired set (e.g., feasible region, Pareto front, or convex hull), sharpened at each step by incorporating valid inequalities (“cuts”) derived from convex duality, subgradients, or quadratic minorants, until a prescribed approximation criterion is met. The approach exploits convexity and duality in generating supporting hyperplanes/tangent cones, with extensions to weak convexity, nonsmoothness, and distributed computation. Modern algorithmic developments address parameter selection, warm-start strategies, strong cutting planes in mixed-integer quadratic/semidefinite settings, and robust/distributed protocols for ellipsoidal approximation.

1. Fundamental Principles of Outer Approximation

Outer approximation constructs an inclusion-wise decreasing sequence of outer sets $\{\mathcal{O}^k\}$ (polyhedral, quadratic, or otherwise convex) that contain the feasible set, upper image, or optimal set of the original problem. At each iteration, current relaxations are interrogated at candidate points (typically extreme points/vertices), which are tested for feasibility or proximity; infeasibility or large deviation triggers the derivation of a valid inequality (cut) that strictly separates the offending point from the true set. This results in an updated relaxation $\mathcal{O}^{k+1} \subset \mathcal{O}^{k}$, and the process continues until a measure, often the Hausdorff distance $d_H(\mathcal{O}^k, \mathcal{I}^k)$ to an inner approximation $\mathcal{I}^k$, falls below a tolerance $\varepsilon$.

In convex vector optimization, scalarizations (Pascoletti–Serafini, norm-minimization, etc.) or subdifferential-based supporting hyperplanes are used to generate cuts (Löhne et al., 2013, Dörfler et al., 2020, Ararat et al., 2021, Keskin et al., 2021). In convex MINLP, the method alternates between fixing integer variables (or block assignments) and solving continuous convex relaxations, then accumulates valid linear or quadratic cuts in a master MILP or MIQCP (Wei et al., 2015, Wei et al., 4 Feb 2026). Extensions include quadratic cuts for weakly convex constraints (Bednarczuk et al., 2024), tight convex quadratic underestimators for deterministic global optimization (Strahl et al., 2024), and spectral cuts for semidefinite programs (Roux et al., 23 Jun 2025).

Formally, core algorithmic ingredients comprise:

• Generation of supporting cuts via subproblems (scalarized convex, feasibility, dual, or quadratic minorant programs).
• Update of the outer relaxation by intersecting with the new cut.
• Use of inner approximations (convex hull of collected feasible images/minimizers) for error estimation and termination.
• Primal–dual or distributed variants, depending on problem structure.

2. Classical and Modern Methodological Variants

a. Convex Vector Optimization and Benson-type Algorithms

Benson's outer approximation forms outer and inner polyhedral approximations of the upper image $\mathcal{P} = \operatorname{cl}(F[S] + C)$, successively refining the approximation by cutting off vertices maximally distant from the current inner set (Dörfler et al., 2020). Scalarizations (e.g., $(P_2(v,c))$ in (Dörfler et al., 2020), norm-minimization in (Ararat et al., 2021), or Pascoletti–Serafini in (Keskin et al., 2021)) provide the supporting hyperplanes. Key innovations include optimal vertex-selection rules that maximize Hausdorff error reduction per iteration, strong duality-based cut generation, and practical stopping criteria rooted in the Hausdorff metric. Projective geometric reformulations eliminate combinatorial vertex explosion in high-dimensional Pareto frontiers (Burton et al., 2010).

b. Convex and Nonsmooth MINLP: Outer Approximation and Variants

The OA method for MINLP alternates between solving NLP subproblems with fixed integer variables and a discrete master problem (MILP or MIQP) that aggregates linearizations at previously visited points (Wei et al., 2015). For nonsmooth or weakly convex problems, subgradients satisfying KKT or other convexity-based optimality conditions replace classical gradients (Wei et al., 4 Feb 2026, Wei et al., 2015, Bednarczuk et al., 2024). Stronger MILP relaxations are achieved by careful construction of cuts, e.g., using scaling parameters ($\rho_j$), alternative feasibility subproblem formulations, or subgradient selection strategies linked to the structure of the constraints (Wei et al., 4 Feb 2026).

Modern practical enhancements include:

• Warm-start initialization for parameterized problem sequences, leading to potential one-iteration solves within subregions of parametric stability (Tamm et al., 11 Jul 2025).
• Event-triggered switching to quadratic cuts when progress stalls (as in DiPOA (Olama et al., 2022)).
• Distributed/partially distributed frameworks