---
title: Fischer–Burmeister Function Overview
url: https://www.emergentmind.com/topics/fischer-burmeister-function
type: topic
---

# Fischer–Burmeister Function Overview

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 $(\cos\theta-1,\,\sin\theta-1)$ for $\theta\in[0,2\pi]$ [1809.09920].

## 2. Smoothing and Regularization

To address the non-differentiability at $(0,0)$, various smooth approximations are used. The canonical smooth FB approximation is
\[
\phi_{\varepsilon}(a, b) = \sqrt{a^2 + b^2 + 2\varepsilon} - a - b
\]
with $\varepsilon>0$ [1805.05801, 1809.09920], and sometimes
\[
\Phi_\varepsilon(a, b) = a + b - \sqrt{a^2 + b^2 + \varepsilon^2}
\]
[1807.03214]. In either case, $\phi_\varepsilon$ is $C^\infty$ on $\mathbb{R}^2$ and converges pointwise (and uniformly on bounded sets) to $\phi_{\mathrm{FB}}$ as $\varepsilon\to 0$. 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 [1807.03214].

## 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
\[
F_\varepsilon(x) = 0,
\]
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 $\phi_{\mathrm{FB}}$ at each iteration, enabling superlinear or quadratic convergence under standard assumptions [1805.05801, 1807.03214]. For smooth $\phi_\varepsilon$, 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 $\varepsilon$ over iterations.

### Penalty and Squared Residuals

Penalty methods for enforcing NCP constraints utilize a least-squares penalty with the FB function:
\[
F(u,v) = \tfrac{1}{2}\int_\Omega \phi(u(x),v(x))^2\,dx.
\]
The solution of the penalized problem converges to a solution of the original NCP as the penalty parameter increases or the smoothing parameter vanishes [1809.09920].

### 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:
\[
\mathcal{L}_{\mathrm{KKT}} = w_{\mathrm{KKT}}
\frac{1}{N_c}\sum_{i=1}^{N_c}\Big[\phi_{\mathrm{FB}}(\tilde{g}_n(x_i), -\tilde{p}_n(x_i))\Big]^2,
\]
where $g_n$ is the normal gap and $p_n$ the normal pressure. The squared loss is $C^1$ even at $0$, making it suitable for gradient-based optimization frameworks and autodifferentiation [2412.09022].

### 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 [1402.6909].

## 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:
\[
\Theta_i(u) = \phi_{\mathrm{FB}}(a_i(u), b_i(u)) = 0
\]
with $a_i(u)$, $b_i(u)$ depending on saturations and pressures. Replacing traditional primary variable switching methods, this approach enables robust simulation with larger time steps and improved nonlinear convergence [1805.05801].

### 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):
\[
\phi_{\mathrm{FB}}(g_n, p_n) = 0, \quad \phi_{\mathrm{FB}}(\mu|T_n| - |T_t|, u_t) = 0
\]
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 [1402.6909, 2412.09022].

### Optimal Control with Complementarity

The FB function and its penalty liftings are fundamental for recasting optimization problems with control constraints $u(x)\ge0\perp v(x)\ge0$ a.e. into unconstrained penalty or smooth root-finding formulations, supporting theory for existence, convergence, and first-order stationary points in function spaces [1809.09920].

### 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 [1807.03214].

## 5. Analytical and Numerical Properties

### Theoretical Convergence

For all $\varepsilon>0$, 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 [1807.03214]. 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 [1809.09920]. In canonical duality applications, global minima of the FB-penalized energy coincide with the solution set of KKT systems for frictional contact [1402.6909].

### 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) [1805.05801]. 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 [2412.09022].

### Implementation Features

The FB function’s partial derivatives are elementary to code for autodiff:
\[
\frac{\partial \phi_{\mathrm{FB}}}{\partial a} = 1 - \frac{a}{\sqrt{a^2 + b^2}},\quad
\frac{\partial \phi_{\mathrm{FB}}}{\partial b} = 1 - \frac{b}{\sqrt{a^2 + b^2}}
\]
[2412.09022]. 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 [2412.09022, 1809.09920]. Smoothing further improves Jacobian structure for large-scale or ill-conditioned problems [1805.05801, 1807.03214]. 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 [1402.6909]. 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)| $\sqrt{a^2 + b^2} - a - b$                          | Lipschitz, C¹ except at $(0,0)$        |
| Smoothed (symmetric)          | $\sqrt{a^2 + b^2 + 2\varepsilon} - a - b$           | $C^\infty$, gradient everywhere        |
| Smoothed (asymmetric)         | $a + b - \sqrt{a^2 + b^2 + \varepsilon^2}$          | $C^\infty$, 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 [1805.05801, 2412.09022, 1402.6909, 1809.09920, 1807.03214].

Source: https://www.emergentmind.com/topics/fischer-burmeister-function