Convexification: Transforming Nonconvex Problems

• Convexification is a method that transforms nonconvex inverse and optimization problems into strictly convex forms using Carleman weight functions and tailored relaxations.
• It employs change of variables, series expansions, and weighted cost functionals to overcome spurious local minima and enable global optimization.
• This framework is applied in PDE coefficient recovery, travel-time tomography, and nonconvex programming, offering provable convergence and stability even under noise.

A convexification method is a numerical and analytical framework for transforming inherently nonconvex (and often ill-posed) inverse problems or nonconvex optimization problems into forms that are globally strictly convex over prescribed admissible sets. By leveraging analytic tools such as Carleman weight functions and carefully constructed convex relaxations (e.g., αBB relaxations, McCormick envelopes), convexification enables the application of global optimization algorithms (notably, gradient descent without reliance on sophisticated initialization) to problems that are otherwise plagued by spurious local minima, ensuring uniqueness of the solution within the admissible region. Convexification methods have seen wide adoption in coefficient inverse problems for partial differential equations (PDEs), nonlinear programming, mean field games, travel-time tomography, and fractional and polynomial programming.

1. Principles of Convexification and Underlying Operators

Convexification centers on embedding the original nonconvex problem into a cost functional that gains strict convexity through the addition of a specially tailored weight—usually a Carleman weight function. In the context of coefficient inverse problems for PDEs, this weight is constructed via a Carleman estimate for the forward operator (elliptic, parabolic, hyperbolic, or transport), granting the cost functional a curvature term that dominates all nonconvexities when the Carleman parameter is sufficiently large. The typical pipeline comprises:

• Change of variables: Often via a logarithmic (or projective) transformation to linearize the unknown coefficient or ratio structure.
• Series expansion: Truncation in an appropriate orthonormal basis (e.g., Fourier, Gram-Schmidt, polynomial) for auxiliary or parameter variables, reducing dimensionality and facilitating finite-dimensional approximation (Klibanov et al., 2022, Smirnov et al., 2020).
• Weighted cost functional: Construction of a least-squares-type or Tikhonov-type functional involving the Carleman weight, usually of the form

$$J_{\lambda}(u) = \int_{\Omega} |F(u)|^2\, \varphi_{\lambda}(x)\,dx + \beta \|u\|^2_{H^k},$$

where $\varphi_{\lambda}$ is the Carleman weight and $F(u)$ the PDE or optimization residual.

• Convexification theorem: Proof that for sufficiently large Carleman parameter (e.g., $\lambda$), the functional $J_{\lambda}$ is globally strictly convex over an admissible ball in a Hilbert or Banach space (Klibanov et al., 2022, Klibanov et al., 2020).

Convexification techniques also encompass relaxations of nonconvex optimization problems, using piecewise αBB relaxations, McCormick envelopes, or moment-hull/SDP-based relaxations to transform the feasible set or objective into a sequence of convex surrogates (Zhu et al., 2022, He et al., 2023).

2. Carleman Weight Functions and Strict Convexity

The Carleman estimate is the analytic backbone enabling convexification in PDE-based problems. For a differential operator $L$, a Carleman estimate provides a lower bound of the type

$$\|L u\, e^{\lambda \psi}\|_{L^2}^2 \geq C \lambda \|\nabla u\, e^{\lambda \psi}\|_{L^2}^2 + C \lambda^3 \|u\, e^{\lambda \psi}\|_{L^2}^2 - \text{boundary terms},$$

with a spatial (or spatiotemporal) weight $\psi$ and Carleman parameter $\lambda \gg 1$.

By employing a Carleman-weighted least squares functional, one inherits this strict convexity:

• For any two admissible functions $u_1, u_2$,

$$J_\lambda(u_2) - J_\lambda(u_1) - \langle J'_\lambda(u_1), u_2-u_1 \rangle \geq C\, \lambda\, \|u_2-u_1\|^2_{H^1_w} + \cdots,$$

where $H^1_w$ denotes an appropriate (weighted) Sobolev space (Klibanov et al., 2022, Smirnov et al., 2020).

The dominance of the positive $\lambda$-weighted terms in the cost gives rise to global strict convexity provided $\lambda$ is chosen large enough to absorb all negative or nonlinear contributions, as proved via a Taylor expansion and application of the Carleman estimate.

3. Piecewise and Parametric Convexification in Nonconvex Programming

For nonconvex optimization, convexification can proceed through the construction of surrogate problems via box partitioning and convex underestimators. The typical structure involves:

• αBB relaxation: For a box $X=[a,b]$, form a convex relaxation

$$F_X^\alpha(x) = f(x) + \sum_{i=1}^n \alpha_i (a_i-x_i)(b_i-x_i)$$

with $\alpha_i$ set to cover the minimum eigenvalue deficit of $\nabla^2 f$ (Zhu et al., 2022).

• Subdivision and selection: The feasible region is adaptively partitioned into sub-boxes, classifying them as convex or non-convex based on the local convexity certificate, and further splitting only non-convex regions. Solution sets for each convexified subproblem are accumulated and combined into a piecewise-approximate global solution set.
• Approximation properties: There are nested inclusion guarantees relating the piecewise convexification solution set to the global set of ε-minimizers as the box diameters shrink, providing both lower and upper bounds on the global minimizer set (Zhu et al., 2022, Zhu et al., 2022).

Parametric convexification is also prevalent in fractional and polynomial optimization, where projective (Charnes–Cooper) and moment-hull approaches provide exact or hierarchically improving convex relaxations (He et al., 2023).

4. Global Convergence, Stability, and Algorithm Design

The strict convexity established via Carleman weights or convex relaxations enables robust application of global optimization algorithms:

• Gradient descent/gradient projection: Converges from any initialization, with a geometric rate determined by the convexity modulus and Lipschitz continuity of the gradient (Klibanov et al., 2020, Klibanov et al., 2022).
• Global convergence theorems: For noisy data, one obtains stability estimates showing that the minimizer (or sequence of minimizers under data/model perturbation) remains within $O(\delta)$ or $O(\delta^\alpha)$ of the true solution, where $\delta$ is the noise level (Smirnov et al., 2020, Klibanov et al., 2021, Klibanov et al., 2022).
• Finite termination in nonlinear programming: Piecewise convexification techniques in nonconvex programming terminate in finitely many steps with an ε-error bound, provided all convex relaxations and subdivisions are constructed according to prescribed tolerances (Zhu et al., 2022).

The table below summarizes representative convexification methods:

Application	Convexification Mechanism	Reference
PDE coefficient inverse	Carleman-weighted functional	(Klibanov et al., 2022)
Nonconvex box optimization	Piecewise αBB relaxations	(Zhu et al., 2022)
MINLP	AVM/McCormick envelopes	(Peng et al., 30 Jul 2024)
Fractional programming	Projective/moment-hull	(He et al., 2023)

5. Applications and Impact Across Domains

Convexification is central in a range of research areas:

• Coefficient inverse problems for PDEs: Convexification has produced the first globally convergent algorithms for coefficient inverse problems in elliptic, parabolic, hyperbolic, and kinetic transport equations—yielding solutions unaffected by initial guess and provably convergent under weak data conditions (Klibanov et al., 2022, Klibanov et al., 2020, Smirnov et al., 2020).
• Seismic and travel-time tomography: Convexification for the eikonal equation enables robust recovery of refractive index from travel-time data in cylindrical domains, with strict convexity and quantitative stability in the weighted functional (Klibanov et al., 21 Sep 2024).
• Nonconvex (multi-objective) optimization: Piecewise convexification provides globally convergent approximations to sets of efficient (Pareto-optimal) solutions in box-constrained multi-objective settings (Zhu et al., 2022).
• Hamilton–Jacobi and mean-field games: Convexification with Carleman weights yields globally strictly convex cost functionals, making possible the direct numerical approximation of viscosity solutions and global-in-time recovery of terminal states (Klibanov et al., 2021, Klibanov et al., 2023).
• Fractional and polynomial programming: Convexification via projective variable transformations, moment-hull or copositive programming yields tight relaxations and finite hierarchy convergence guarantees (He et al., 2023).

Notably, these methodologies have demonstrated practical impact in large-scale experimental data recovery (dielectric imaging, imaging of moving targets), global optimization in mixed-integer nonlinear programming (MINLP), and provide templates for extending Carleman-weighted convexification to new inverse and optimization problems.

6. Implementation, Limitations, and Prospective Directions

Convexification methods are implemented in diverse frameworks—ranging from MATLAB/unconstrained optimizers for PDE CIPs, to Pyomo–MindtPy (with BARON/Gurobi/IPOPT) for MINLP, and bespoke subdivision solvers for piecewise convexification in nonlinear programming and multi-objective settings (Peng et al., 30 Jul 2024, Zhu et al., 2022, Zhu et al., 2022). Key implementation observations include:

• Parameter selection: While theory dictates the need for large Carleman parameters for strict convexity, practical computations indicate moderate parameter values suffice.
• Handling of noise and model errors: Stability and convergence results are robust to moderate noise in the data, with solution accuracy scaling with the noise magnitude and regularization strength.
• Complexity and scalability: Piecewise or subdivision-based convexification is subject to combinatorial growth in problem dimensionality—motivating adaptive and focused partitioning.
• Current limitations: Some theoretical gaps remain in connecting truncations (series expansions) or fully discrete approximations to exact solutions in the limit (e.g., (Klibanov et al., 2020) establishes only $L^2$ convergence of residuals, and cannot guarantee convergence of minimizers to the true solution, due to severe ill-posedness).

The extension of convexification mechanisms to inverse problems for new classes of nonlinear PDEs, more general optimization domains, and broader multi-term nonconvex objectives continues to be an area of active development. Current research focuses on adaptive basis selection, optimal design of Carleman weights, and hybridization with global optimization heuristics for large-scale or mixed-integer settings.