Fischer–Burmeister Function Overview

• Fischer–Burmeister function is a nonlinear complementarity function that recasts complementarity constraints as a single equation, offering global Lipschitz continuity and semi-smooth properties.
• Smoothing and regularization techniques, such as adding an epsilon parameter, ensure differentiability and improve convergence in semi-smooth Newton and penalty-based numerical methods.
• The function finds practical applications in optimal control, contact mechanics, multiphase flow, and physics-informed neural networks, enabling robust and efficient constraint handling.

The Fischer–Burmeister function is a prototypical nonlinear complementarity problem (NCP) function that recasts complementarity constraints in terms of a single equation. It plays a central role in reformulations of the Karush–Kuhn–Tucker (KKT) conditions for a wide range of optimization, complementarity, and variational inequality problems, as well as in modern numerical methods involving semi-smooth Newton algorithms, penalty and smoothing strategies, and machine learning approaches for physics-constrained systems. The function is widely adopted in mathematical programming, optimal control, contact mechanics, and multiphase flow simulation due to its structural properties and compatibility with gradient-based solvers.

1. Definition and Core Properties

The Fischer–Burmeister (FB) function, denoted $\phi_{\mathrm{FB}}: \mathbb{R}^2 \to \mathbb{R}$, is defined by

$$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$

It is classified as a C-function for NCPs, meaning

$$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$

The function is continuous and globally Lipschitz on $\mathbb{R}^2$ (with constant $\leq 2$). $\phi_{\mathrm{FB}}$ is differentiable except at the origin $(0,0)$, where its subdifferential is set-valued but always contains the zero vector. Away from $(0,0)$, the partial derivatives are

$$\frac{\partial\phi_{\mathrm{FB}}}{\partial a}(a, b) = \frac{a}{\sqrt{a^2 + b^2}} - 1,\qquad \frac{\partial\phi_{\mathrm{FB}}}{\partial b}(a, b) = \frac{b}{\sqrt{a^2 + b^2}} - 1.$$

The Clarke subdifferential at $(0, 0)$ is the convex hull of $$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$0 for $$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$1 (Clason et al., 2018).

2. Smoothing and Regularization

To address the non-differentiability at $$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$2, various smooth approximations are used. The canonical smooth FB approximation is

$$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$3

with $$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$4 (Bui et al., 2018, Clason et al., 2018), and sometimes

$$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$5

(Liao-McPherson et al., 2018). In either case, $$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$6 is $$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$7 on $$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$8 and converges pointwise (and uniformly on bounded sets) to $$\phi_{\mathrm{FB}}(a, b) = \sqrt{a^2 + b^2}\; -\; a\; -\; b.$$9 as $$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$0. The smoothing parameter ensures that the Jacobian is well-defined everywhere, improving the behavior of Newton-type solvers and globalizing convergence. Regularization strategies may further improve conditioning, for example by augmenting the system’s linearization with a term proportional to the primal and dual variables (Liao-McPherson et al., 2018).

3. Algorithmic Roles

Semi-Smooth Newton and Smoothing-Newton Methods

The FB function enables the reformulation of KKT/complementarity conditions into a semi-smooth equation system. In multiphase flow and quadratic programming, these equations take the form

$$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$1

where the FB function is applied component-wise to pairs of primal and dual variables (or their residuals). Semi-smooth Newton methods operate using elements of the B-subdifferential of $$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$2 at each iteration, enabling superlinear or quadratic convergence under standard assumptions (Bui et al., 2018, Liao-McPherson et al., 2018). For smooth $$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$3, the usual Newton derivative is applied. Typical algorithmic steps include:

1. Evaluate the FB (or smoothed FB) residual.
2. Form (generalized) Jacobian or smoothed derivative.
3. Solve the Newton step.
4. (If smoothed) Reduce $$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$4 over iterations.

Penalty and Squared Residuals

Penalty methods for enforcing NCP constraints utilize a least-squares penalty with the FB function: $$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$5 The solution of the penalized problem converges to a solution of the original NCP as the penalty parameter increases or the smoothing parameter vanishes (Clason et al., 2018).

Physics-Informed Neural Networks (PINNs)

In PINNs for contact mechanics, the FB function is used to encode the complementarity KKT conditions as a soft loss: $$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$6 where $$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$7 is the normal gap and $$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$8 the normal pressure. The squared loss is $$\phi_{\mathrm{FB}}(a, b)=0 \iff a\ge 0,\; b\ge 0,\; ab=0.$$9 even at $\mathbb{R}^2$0, making it suitable for gradient-based optimization frameworks and autodifferentiation (Sahin et al., 2024).

Canonical Duality and Global Optimization

The FB function provides a merit function to reformulate variational inequalities such as frictional contact with KKT constraints into a global optimization problem. Canonical dual transformation introduces auxiliary dual variables, recasting the original nonconvex problem as a saddle-point problem with strong duality under regularity assumptions (Latorre et al., 2014).

4. Applications Across Research Domains

Multiphase Flow in Porous Media

The NCP formulation for compositional two-phase flow with phase transitions employs the FB function (or its smooth variant) to encode cellwise phase appearance/disappearance: $\mathbb{R}^2$1 with $\mathbb{R}^2$2, $\mathbb{R}^2$3 depending on saturations and pressures. Replacing traditional primary variable switching methods, this approach enables robust simulation with larger time steps and improved nonlinear convergence (Bui et al., 2018).

Contact Mechanics and Friction

Contact boundary problems in elasticity utilize the FB function for pointwise normal and tangential complementarity (contact gap and pressure, slip and friction force): $\mathbb{R}^2$4 allowing collapsing of inequality and orthogonality conditions into single equations per contact node. In both classical variational and deep learning/PINN frameworks, the FB penalty predicts both constraint satisfaction and accurate complementarity (Latorre et al., 2014, Sahin et al., 2024).

Optimal Control with Complementarity

The FB function and its penalty liftings are fundamental for recasting optimization problems with control constraints $\mathbb{R}^2$5 a.e. into unconstrained penalty or smooth root-finding formulations, supporting theory for existence, convergence, and first-order stationary points in function spaces (Clason et al., 2018).

Quadratic Programming and Embedded Optimization

Modern real-time optimization, especially in model predictive control, leverages the FB function to express the QP KKT conditions as a root-finding problem solvable by a globalized, regularized, and smoothed Newton scheme. Warm-start ability, robust convergence, and real-time suitability have been demonstrated in hardware-in-the-loop experiments for embedded model predictive control (Liao-McPherson et al., 2018).

5. Analytical and Numerical Properties

Theoretical Convergence

For all $\mathbb{R}^2$6, semismooth Newton methods equipped with the FB (or its smoothing) enjoy local quadratic convergence under LICQ and standard KKT regularity. The regularized Jacobian remains nonsingular even in degenerate situations (Liao-McPherson et al., 2018). The penalty approach with FB (in the control context) ensures convergence of global minimizers of the penalized problems to solutions of the original complementarity-constrained problem as the penalty parameter grows (Clason et al., 2018). In canonical duality applications, global minima of the FB-penalized energy coincide with the solution set of KKT systems for frictional contact (Latorre et al., 2014).

Numerical Robustness

Empirical studies across applications reveal that the smoothed FB function outperforms min-based or other complementarity functions in terms of robustness (fewer failed time steps), efficiency (reduced Newton or optimization iterations), and scalability (efficient parallelization, improved preconditioning due to smoothing) (Bui et al., 2018). For PINNs, use of the squared FB loss yields mean-squared errors at KKT satisfaction points close to machine accuracy without optimizer instability, permitting simple penalty tuning (Sahin et al., 2024).

Implementation Features

The FB function’s partial derivatives are elementary to code for autodiff: $\mathbb{R}^2$7 (Sahin et al., 2024). Smoothed variants inherit higher regularity, further enhancing computational stability and differentiability needed in gradient-based optimizers.

6. Comparative Perspectives and Trade-offs

The FB function offers advantages over minimum-based, hinge, or piecewise NCP functions due to its global Lipschitz continuity, absence of directional kinks, and unique penalty parameterization. Its squared residual is C¹ (even at the origin), which empirically yields more stable optimization both in classical and modern neural frameworks (Sahin et al., 2024, Clason et al., 2018). Smoothing further improves Jacobian structure for large-scale or ill-conditioned problems (Bui et al., 2018, Liao-McPherson et al., 2018). However, the FB function is nonconvex in (a, b), limiting its direct applicability as a global convex penalty. In nonconvex programs these properties require either global optimization or regularization strategies to avoid spurious solutions (Latorre et al., 2014). The use of the FB function is typically preferred for its balance of mathematical rigor, computational tractability, and amenability to both derivative-based and semismooth solution algorithms.

7. Summary Table: Fischer–Burmeister Function Key Variants

Variant	Definition	Key Properties
Fischer–Burmeister (classical)	$\mathbb{R}^2$8	Lipschitz, C¹ except at $\mathbb{R}^2$9
Smoothed (symmetric)	$\leq 2$0	$\leq 2$1, gradient everywhere
Smoothed (asymmetric)	$\leq 2$2	$\leq 2$3, used in QP solvers

These formulations underpin advanced approaches in nonlinear and variational optimization, complementarity-constrained control, PDE-constrained contact mechanics, and physics-informed machine learning, demonstrating both theoretical soundness and wide empirical success (Bui et al., 2018, Sahin et al., 2024, Latorre et al., 2014, Clason et al., 2018, Liao-McPherson et al., 2018).