FICA: Faster Inner Convex Approximation

Updated 22 July 2025

FICA is a framework of algorithms that efficiently generates inner convex approximations of complex feasible regions in high-dimensional optimization and control problems.
It leverages methods such as excessive gap minimization and grid-agnostic skeleton approaches to achieve faster convergence and reduced computational cost.
FICA has broad applications in robust power dispatch, multiobjective optimization, and control systems, offering scalable and reliable solution strategies.

Faster Inner Convex Approximation (FICA) refers to a family of algorithmic frameworks and methodologies for efficiently producing inner convex approximations of sets or feasible regions that appear in high-dimensional optimization, geometric, and control-theoretic problems. These methods are distinguished by their emphasis on both computational tractability and the ability to “tightly” approximate complicated sets from the inside, thereby enabling reliable optimization or certification of solutions under various problem structures, including nonsmoothness, nonconvexity, or uncertainty. FICA finds application in areas such as multiobjective optimization, computational geometry, robust control, robust and chance-constrained optimization, variational problems with convexity constraints, and learning convex functions.

1. Theoretical Foundations and Algorithmic Paradigms

Early approaches to convex inner approximation frequently relied on geometric arguments—most notably, incremental construction of “coresets” or greedy selection procedures, as in the Minimum Enclosing Ball (MEB) and Minimum Enclosing Convex Polytope (MECP) problems. Faster algorithms, such as those based on the excessive gap technique (0909.1062), recast these geometric covering or containment problems as non-smooth convex optimization with explicit duality. The excessive gap framework of Nesterov is specialized to generate primal–dual iterates such that the approximation error (quantified via the duality gap $J(w_k) - D(u_k)$ ) converges at a rate $O(1/k^2)$ , requiring only $O(Q/\sqrt{\epsilon})$ iterations to reach an $\epsilon$ -accurate solution, where $Q$ bounds the norm of data points. This rate is markedly better than that of greedy coreset methods, which have $O(1/\epsilon)$ dependence.

Similar primal–dual frameworks underpin FICA approaches for convex vector optimization (Löhne et al., 2013), where sequences of polyhedral approximations (outer and inner) can be constructed in parallel, with each “cut” improving both the internal and external approximations. Here, the $\epsilon$ -solution concept is central: the inner approximation (the convex hull of representative efficient points) is guaranteed to be “ $\epsilon$ -close” to the true Pareto front in the sense of set containment.

Grid-agnostic skeleton methods and plane-separating inner approximation algorithms (Nemesch et al., 28 Apr 2025, Csirmaz, 2018) further accelerate polyhedral inner approximations in high-dimensional multiobjective settings by eliminating the need for fine tessellation of the objective or weight space and by focusing on facets where the current approximation is most deficient.

2. Methodologies for Convex Inner Approximation

Methodological advances in FICA span a range of application domains and set descriptions:

Semialgebraic Sets: For sets defined by multivariate polynomials, inner convex approximation can be obtained by iteratively examining curvature along the boundaries (via gradients and Hessians) and introducing affine cuts at points of negative curvature, thereby “clipping” nonconvex portions while preserving convex boundaries (1104.2679). The polynomial optimization steps involved are efficiently managed with LMI relaxations (e.g., via Gloptipoly 3 and SDP solvers).
Variational Problems and Convexity Constraints: In problems with functional convexity constraints (e.g., convex regression, convex envelope computation), the infinite-convexity constraint is replaced by finitely many directional tests (Oberman, 2011). For a finite grid of directions, second derivatives are required to be nonnegative, converting the global convexity condition into linear inequalities amenable to LP/QP solvers.
Chance-Constrained and Distributionally Robust Optimization: Convex inner approximations of chance constraints with right- or left-hand-side uncertainty (RHS/LHS) are realized by replacing nonconvex constraints over uncertainty supports by efficiently solvable linear or convex programs using valid inequalities and order-statistical arguments (Zhou et al., 17 Dec 2024, Zhou et al., 23 Jun 2025). In problems where constraints have one-dimensional structure (e.g., linear in a single uncertainty), FICA achieves dramatic reductions in constraint count and solution time while maintaining the optimality guarantee of earlier CVaR-based methods.
Polyhedral Multiobjective Approximation: Grid-agnostic skeleton algorithms—derived from polyhedral geometry—construct inner approximations using separation oracles and iteratively updating convex hulls, obviating the need for grid-based enumeration and thus improving both running time and storage requirements (Nemesch et al., 28 Apr 2025).

3. Acceleration and Computational Efficiency

A recurring theme in the development of FICA is the leveraging of problem structure—such as low-dimensional uncertainty in parametric constraints, convexity along selected directions, or favorable duality relationships—to realize significant speedup. Notable strategies include:

Dynamic Regularization: In convex bi-level optimization, dynamically tuning a regularization parameter (e.g., via FBi-PG) allows the inner convex problem to be solved with $O(1/k^2)$ complexity while ensuring convergence of both inner and outer objectives (Merchav et al., 30 Jul 2024).
Directionally Reduced Constraints: Enforcing directional rather than full orthant convexity translates to a linear (rather than quadratic or higher) number of constraints; this brings multiple orders-of-magnitude speedups in variational and regression problems (Oberman, 2011).
Valid Inequalities and One-Dimensional Structure: For Wasserstein chance constraints, inner approximations that exploit one-dimensional coupling between variables and uncertainty achieve up to 500 $\times$ speedup for large-scale grid dispatch problems (Zhou et al., 23 Jun 2025).
Efficient Data Structures: For large-scale search and hybrid inner product queries in machine learning, the use of cache-sorted indices, quantization, and SIMD-friendly lookup tables enables efficient approximation of similarities in high-dimensional hybrid spaces (Wu et al., 2019).

4. Applications and Practical Impact

FICA methodologies have enabled advances across varied application areas:

Power Systems and Robust Dispatch: Convex inner approximations allow secure, tractable, and scalable scheduling of generation resources under uncertainty, with guarantees against voltage or current violations in distribution networks (Nguyen et al., 2017, Nazir et al., 2019, Zhou et al., 23 Jun 2025). In bilevel and multi-stage settings with coupled decision-uncertainty constraints, FICA facilitates solution of previously intractable models.
Multiobjective and Vector Optimization: Inner convex approximations allow simultaneous, polynomial-time construction of representative sets which approximate the complete Pareto front, including for large-scale assignment, knapsack, and travelling salesman problems (Helfrich et al., 2023, Nemesch et al., 28 Apr 2025).
Control and Motion Planning: In fixed-order controller synthesis, FICA produces conservative but guaranteed-stable inner sets within which controller parameters can be optimized (1104.2679). For robotic trajectory planning, inner convex approximations of collision-free sets (e.g., via “free balls”) guarantee continuous-time collision avoidance and kinodynamic feasibility in nonlinear model predictive control frameworks (Schoels et al., 2019).
Learning and Regression under Convexity: Efficient convex regression, DC-function learning, and metric learning with convex constraints are made tractable on large datasets via ADMM-based FICA schemes, reducing computational cost by up to two orders of magnitude (Siahkamari et al., 2021).

5. Comparative Performance and Limitations

FICA approaches uniformly improve over classical methods such as greedy coresets, full grid enumerations, and fixed-parameter regularizations. Improvements are most marked when underlying problem structure (e.g., sparsity, dimensionality, one-dimensional uncertainty) is exploited.

In chance constraint optimization, FICA methods based on the Strengthened and Faster Linear Approximation (SFLA) (Zhou et al., 17 Dec 2024) and LHS-structure (Zhou et al., 23 Jun 2025) demonstrate 10× to 500× computational speedups over exact or CVaR-based approximations, with typical optimality gaps below 1%. Skeleton-based grid-agnostic algorithms in multiobjective optimization outperform grid-based dual Benson and OAA-type methods in both solution set size and running time.

Limitations are context-dependent. Conservativeness remains a consideration in highly constrained systems, especially with high risk levels or limited sample supports in chance-constrained optimization. Efficient exploitation of one-dimensional structure is not universally possible, and in applications lacking exploitable low-dimensional representations or ordering arguments, the standard CVaR or LA approximations may remain necessary.

6. Directions for Future Research

Key future research topics in FICA include:

Adaptive and Data-Driven Hyperparameter Tuning: Developing automated schemes to select grid densities, valid inequality strengths, or regularization parameters as a function of instance data.
Scenario Reduction and Scalability: Extending scenario reduction techniques to inner approximation settings while maintaining strong guarantees.
Extending to Nonconvex and Multistage Settings: Integrating FICA within iterative or nested schemes to address nonconvexity or recourse in multistage settings, especially in grid dispatch and logistics.
Quantifying Conservativeness and Robustness: Systematic paper of approximation quality in the presence of model misspecification and its impact on solution quality.
Algorithmic Implementation and Practical Optimization: Further development of algorithmic toolkits (efficient oracle calls, representation conversion, and polyhedral manipulation) supporting rapid prototyping of FICA methods across domains.

7. Key Concepts and Mathematical Formulations

FICA frameworks employ a range of mathematical tools, including:

Excessive gap minimization for $O(1/\sqrt{\epsilon})$ convergence (0909.1062).
Directionally-enforced convexity by linear inequalities (Oberman, 2011).
Polyhedral inner approximation via separation oracles and convex hull updating (Nemesch et al., 28 Apr 2025, Csirmaz, 2018).
SFLA and order-statistics for uncertainty in chance constraints (Zhou et al., 17 Dec 2024, Zhou et al., 23 Jun 2025).
Dynamic composite regularization for accelerated algorithms (Merchav et al., 30 Jul 2024).
Scale-invariant volume minimization in polynomial sublevel approximations of semialgebraic sets (Guthrie, 2022).

The common structural form of convex inner approximations can often be cast as:

$\text{Find } S_{\text{in}} \subseteq S \text{, with } S_{\text{in}} \text{ convex, maximizing certain coverage, minimized conservativeness, or computational efficiency,}$

subject to tractable representations, as in

$\min_{w \in Q_1} \left\{ f(w) + \max_{u \in Q_2} \langle Aw, u \rangle - g(u) \right\},$

$\begin{aligned} & \mathcal{A}^{(k+1)} = \operatorname{conv}\{ f(x) : x \in R \} + R, \ & H_{(1+\epsilon)} = \{ z \in \mathbb{R}^d : (1 + \epsilon) w^\top z \geq c \}, \end{aligned}$

where each update incrementally “fills in” the convex approximation until the desired level of approximation is achieved.

These methodological and theoretical advances continue to broaden the scope and impact of FICA, providing efficient and reliable tools for inner convex approximation across increasingly complex real-world optimization problems.