Homotopy-Based Constraint Relaxation

• Homotopy-based constraint relaxation is a method that replaces hard constrained problems with a parameterized family of smoother, tractable surrogate problems.
• It employs continuation techniques where solutions of earlier subproblems serve as warm starts, gradually steering toward the original constrained solution.
• The approach is applied in diverse fields—such as optimization, motion planning, and Bayesian inference—maintaining asymptotic links to the target problem’s structure.

Homotopy-based constraint relaxation denotes a class of methods that replace a hard constrained problem by a parameterized family of easier problems and then trace solutions as the parameter approaches the target regime. In the literature, the relaxed family may deform the feasible set, smooth an indicator or complementarity relation, embed inequalities into a barrier, or replace an overdetermined system by adaptive well-constrained subsystems. The continuation parameter is then decreased or increased—depending on the formulation—while solutions of earlier subproblems are used as warm starts for later ones. This paradigm appears in switching-constrained and complementarity-constrained programming, convex and conic optimization, parameterized polynomial systems, Bayesian inference under equality and inequality constraints, motion planning, sparse recovery, and density-based topology optimization (Kanzow et al., 2018, Klingler et al., 2024, Hauenstein et al., 2017, Duan et al., 2018, Gangl et al., 7 Jan 2026).

1. Parameterized reformulation and continuation principle

A common formal pattern is to replace an original constrained problem by a one-parameter family

$$P_\tau:\quad \min_x f_\tau(x)\quad \text{s.t.}\quad x\in X_\tau,$$

or, at the level of first-order conditions, by a parameterized residual equation

$H(z,\tau)=0.$

The homotopy parameter $\tau$ is chosen so that one endpoint corresponds to an easy problem and the other to the original problem. In switching-constrained optimization, one solves relaxed nonlinear programs $P(t_k)$ with $t_k\downarrow 0$ (Kanzow et al., 2018). In feasible-set homotopies for convex optimization, one solves

$\max_{x\in C_t} f(x), \qquad t\in[0,1],$

with $C_0$ easy and $C_1=C$ the target set (Klingler et al., 2024). In barrier-based conic methods, the family is

$$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$

so increasing $t$ tightens the effect of the barrier relaxation and traces the analytic central path (Dvurechensky et al., 2022). In Bayesian constraint relaxation, the family of posteriors is

$H(z,\tau)=0.$0

with $H(z,\tau)=0.$1 recovering the sharply constrained model and $H(z,\tau)=0.$2 recovering the unconstrained posterior (Duan et al., 2018).

The continuation viewpoint is not restricted to scalar schedules. Probabilistic Homotopy Optimization formulates the homotopy as a map $H(z,\tau)=0.$3 from a multidimensional parameter cube into nonlinear-program parameters, so that each sampled $H(z,\tau)=0.$4 defines a constrained optimization problem $H(z,\tau)=0.$5 (Pardis et al., 2024). Parameter homotopy for polynomial systems instead interpolates problem parameters directly through

$H(z,\tau)=0.$6

and tracks solution paths with predictor-corrector continuation (Hauenstein et al., 2017). The sequential homotopy method for Hilbert-space mathematical programming embeds the original problem into primal-dual proximal-regularized subproblems indexed by a stepsize or regularization parameter (Potschka et al., 2019).

Across these formulations, homotopy-based constraint relaxation is characterized by three elements: a family of surrogate problems with improved regularity or tractability, a path-following mechanism, and a limiting argument that relates terminal points of the path to constrained stationary points or optimizers of the original problem.

2. Relaxation mechanisms

The concrete form of the relaxation varies widely across problem classes. Some methods enlarge or smooth the feasible set; others keep the feasible set implicit and regularize the optimality system.

Setting	Relaxed representation	Limiting regime
MPSC	Quadrant-wise NCP inequalities $H(z,\tau)=0.$7	$H(z,\tau)=0.$8 (Kanzow et al., 2018)
MPCC	Lasry–Lions envelope $H(z,\tau)=0.$9	$\tau$0 (Wang et al., 7 Jul 2025)
Conic/SDP	Barrier objective $\tau$1	$\tau$2 in effect of $\tau$3 bound (Dvurechensky et al., 2022)
Bayesian constraints	Kernelized posterior $\tau$4	$\tau$5 (Duan et al., 2018)
Topology optimization	Primal-dual barrier system with $\tau$6	$\tau$7 (Gangl et al., 7 Jan 2026)
Motion planning	Metric $\tau$8 and geometric heat flow	flow time $\tau$9 (Liu et al., 2019)

In mathematical programs with switching constraints, the Kanzow–Schwartz adaptation uses the smooth NCP function

$P(t_k)$0

and defines four quadrant-specific inequalities $P(t_k)$1 for each switching pair $P(t_k)$2. This yields a relaxed NLP with feasible sets $P(t_k)$3 satisfying $P(t_k)$4, $P(t_k)$5 for $P(t_k)$6, and $P(t_k)$7 (Kanzow et al., 2018). By contrast, Scholtes-type relaxations use a product band $P(t_k)$8, while Steffensen–Ulbrich relaxations smooth absolute values inside four inequalities; these two variants are structurally more coupled in the multipliers they induce (Kanzow et al., 2018).

In MPCCs, the Lasry–Lions double envelope relaxes the indicator of the complementarity set. For $P(t_k)$9,

$t_k\downarrow 0$0

Applied pairwise to complementarity constraints, this produces a globally $t_k\downarrow 0$1 surrogate with Lipschitz gradient and sandwich bounds

$t_k\downarrow 0$2

preserving minima and minimizers while smoothing the nonsmooth indicator (Wang et al., 7 Jul 2025).

Barrier-based relaxations use a different mechanism. In conic optimization, a logarithmically homogeneous self-concordant barrier $t_k\downarrow 0$3 on a cone $t_k\downarrow 0$4 is transported through an affine map $t_k\downarrow 0$5 to obtain $t_k\downarrow 0$6, and the family $t_k\downarrow 0$7 tracks the analytic central path (Dvurechensky et al., 2022). In density-based topology optimization, box constraints on the density are enforced by a primal-dual log-barrier system

$t_k\downarrow 0$8

so strict feasibility is maintained while $t_k\downarrow 0$9 is decreased (Gangl et al., 7 Jan 2026).

Distance-based relaxations appear in Bayesian inference. Equality constraints $\max_{x\in C_t} f(x), \qquad t\in[0,1],$0 are replaced by a Lipschitz residual norm such as $\max_{x\in C_t} f(x), \qquad t\in[0,1],$1, and the sharp indicator is replaced by $\max_{x\in C_t} f(x), \qquad t\in[0,1],$2 (Duan et al., 2018). In motion planning, the relaxation is geometric rather than algebraic: constrained directions are made expensive through a metric $\max_{x\in C_t} f(x), \qquad t\in[0,1],$3, obstacles are encoded by a barrier multiplier $\max_{x\in C_t} f(x), \qquad t\in[0,1],$4, and the final metric is $\max_{x\in C_t} f(x), \qquad t\in[0,1],$5; inadmissible motion is then suppressed by geometric heat flow (Liu et al., 2019).

3. Stationarity notions, regularity, and convergence guarantees

A central issue is the mismatch between classical constraint qualifications and the geometry of the original constrained problem. In mathematical programs with switching constraints, when

$\max_{x\in C_t} f(x), \qquad t\in[0,1],$6

standard MFCQ and often ACQ fail. This motivates weaker stationarity notions. W-stationarity imposes first-order balance with restricted multipliers; M-stationarity adds $\max_{x\in C_t} f(x), \qquad t\in[0,1],$7 at doubly active switching indices; S-stationarity sets both multipliers to zero there. The Kanzow–Schwartz-type relaxation computes M-stationary limit points under MPSC-NNAMCQ, while adapted Scholtes and Steffensen–Ulbrich relaxations generally yield only W-stationarity. The contrast is structural: the quadrant-wise NCP relaxation separates $\max_{x\in C_t} f(x), \qquad t\in[0,1],$8- and $\max_{x\in C_t} f(x), \qquad t\in[0,1],$9-multipliers, whereas product-band and smoothing relaxations do not enforce the decoupled multiplier condition $C_0$0 at double-active indices (Kanzow et al., 2018).

For MPCCs, the Lasry–Lions envelope method uses regular, limiting, and Clarke-type normal cones to the scalar complementarity set $C_0$1. Under approximate first-order stationarity of the smooth subproblems and a basic Clarke CQ, limit points are C-stationary. If the complementarity constraints are well behaved and BCQ$C_0$2 holds, the method yields M-stationary points; an intermediate AM-stationarity concept captures the limiting multiplier sequence before boundedness is established (Wang et al., 7 Jul 2025). The paper also derives a worst-case complexity estimate of order

$C_0$3

for obtaining an $C_0$4-approximate C-stationary point under the stated schedule and first-order inner solves (Wang et al., 7 Jul 2025).

In the sequential homotopy method for mathematical programming in Hilbert spaces, the continuation subproblems are the KKT systems of primal-dual proximal-regularized augmented Lagrangian problems. The associated projected gradient/antigradient flow has equilibria that coincide with critical points of the original constrained problem, and the backward Euler discretization is equivalent to solving the proximal-regularized subproblems. The method is designed so that the regularized problems are always feasible, satisfy a strong CQ, and admit unique multipliers; under the stated assumptions, global flow solutions exist and critical points with emanating descent curves cannot be asymptotically stable equilibria (Potschka et al., 2019).

In convex feasible-set homotopy, the theoretical object is a path of boundary optimizers $C_0$5 satisfying differentiated KKT conditions

$C_0$6

If the homotopy of smooth descriptions is regular and $C_0$7 is strictly quasi-concave on $C_0$8, then the path exists uniquely on $C_0$9, remains on the boundary, and converges to the optimizer of the target convex problem. The invertibility of $C_1=C$0 is supplied by strict quasi-concavity, and correctness follows from uniqueness along the path and compactness at the limit (Klingler et al., 2024).

These results show that homotopy-based constraint relaxation is not a single convergence theorem but a family of limiting arguments. The limit object may be an optimizer, a KKT point, a W/M/S-stationary point, or a probabilistic limit of posterior expectations, depending on how the relaxation interacts with the original constraint geometry.

4. Numerical realization and path-tracking strategies

Most implementations are sequential and warm-started. In switching-constrained optimization, the basic template initializes $C_1=C$1, solves the relaxed NLP $C_1=C$2 to KKT accuracy, computes a feasibility residual, decreases the parameter geometrically by $C_1=C$3, and warm-starts the next subproblem from the current KKT point. The reported practical choices include $C_1=C$4, stopping when $C_1=C$5, and using SNOPT or IPOPT with user-supplied gradients; safeguards include temporarily increasing $C_1=C$6, damping the step, perturbing the start, and monitoring oscillations in relaxed active sets (Kanzow et al., 2018).

ODE-based path following is used when the homotopy is formulated directly on stationarity equations. For convex optimization, the numerical scheme combines an explicit Runge–Kutta 5(4) predictor for

$C_1=C$7

with a Newton-like corrector that reduces the residuals in $C_1=C$8 and the parallelism constraints $C_1=C$9. The step size is adapted so that the endpoint error is at most $$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$0 (Klingler et al., 2024). Parameter homotopy for polynomial systems uses the Davidenko ODE

$$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$1

classical fourth-order Runge–Kutta prediction, Newton correction, adaptive step-size control, and additional adaptive mechanisms for projective patch selection and randomization of overdetermined systems (Hauenstein et al., 2017).

Not all modern variants follow a single continuous path. Probabilistic Homotopy Optimization alternates solve and sample phases over a multidimensional homotopy parameter space. It stores solutions in an Optimization Tree, attempts warm-start solves at previously sampled parameters, and samples new parameters when the ratio of attempted solves to available node-parameter combinations becomes too high. This discrete search in homotopy space is intended to cope with folds, bifurcations, and disconnected solution components that can obstruct classical continuation (Pardis et al., 2024).

Double-loop methods separate homotopy updates from inner constrained minimization. The conditional gradient homotopy method of Dvurechensky, Shtern, and Staudigl applies a Frank–Wolfe inner loop to the barrier-augmented objective $$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$2, with an outer schedule

$$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$3

The inner loop is projection-free and uses a linear minimization oracle; for spectrahedra with trace bounds, each step reduces to a leading-eigenvector computation and hence a rank-one update (Dvurechensky et al., 2022).

Topology optimization couples continuation and interior mechanisms differently. There the homotopy is applied to the primal-dual barrier optimality system, with $$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$4 driving both the original residual and the barrier parameter $$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$5, and Newton correctors solving the resulting block system in $$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$6 (Gangl et al., 7 Jan 2026). In motion planning, the homotopy variable is a flow time $$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$7, and the numerical task is parabolic PDE integration for the geometric heat flow, rather than repeated nonlinear programming solves (Liu et al., 2019).

5. Domains of application

Switching and complementarity problems provide a canonical optimization setting. For mathematical programs with switching constraints, the reported computational experiments cover an either-or constrained problem with 64 starts from $$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$8, a switching-constrained optimal control problem for the heat equation in 2D with 101 time nodes and 1000 random starts in $$x(t)\in\arg\min_x \varphi_t(x),\qquad \varphi_t(x)=t\,g(x)+F(x),$$9, and a semi-continuous portfolio optimization problem with $t$0 over 30 instances. The Kanzow–Schwartz relaxation achieved the best function values in more than 80% of runs in the either-or test, lower objective values in around 70% of runs for the control problem, and best function values in 28/30 portfolio instances, with higher feasibility reliability than non-relaxed NLP solvers (Kanzow et al., 2018). For MPCCs, the Lasry–Lions envelope method was tested on MacMPEC, large-scale bound-constrained quadratic MPCCs, and large-scale linearly constrained quadratic MPCCs, where the reported outcome was a strong balance of robustness, speed, and solution quality, especially under degeneracy (Wang et al., 7 Jul 2025).

Conic and convex optimization supply another major application class. Feasible-set homotopy was developed for compact convex problems with linear objectives and explicit constructions for semidefinite and hyperbolic programming using real-zero polynomials and Renegar smoothing (Klingler et al., 2024). The conditional gradient homotopy method targets convex programs with many simple conic constraints and was specialized to semidefinite relaxations such as MaxCut and fastest mixing Markov chain formulations, where the key computational primitive is a projection-free linear minimization oracle on a spectrahedron (Dvurechensky et al., 2022).

Parameterized algebraic systems and computer vision use homotopy-based relaxation in a different sense. Adaptive patching and adaptive subsystem selection were developed for overdetermined parameterized polynomial systems, with detailed experiments on the 5-point essential matrix problem and a 6-point problem with radial distortion. The reported averages over 100 instances show that coordinate-wise patching plus leverage-score randomization reduced steps per path and runtime relative to fixed patch/fixed randomization, and that an early truncation heuristic further improved performance when many nonreal solutions were present (Hauenstein et al., 2017).

Bayesian inference treats constraints probabilistically. The relaxed posterior

$t$1

was applied to sphere and torus constraints, monotonicity and simplex-type inequalities, and a brain-network latent factor model with near-orthonormality on a Stiefel manifold. In the brain-network study, shrinkage plus near-orthonormality achieved fitted AUC $t$2, prediction AUC $t$3, and ESS/1000 of about $t$4, whereas the completely constrained version had prediction AUC $t$5 and ESS about $t$6 (Duan et al., 2018).

Motion planning uses both continuous and search-based homotopies. The geometric heat-flow method was illustrated on the unicycle, car-like systems, multi-vehicle planning, and a two-link manipulator, where the initial trajectory is only required to avoid obstacles and is deformed into an admissible path by the metric-induced flow (Liu et al., 2019). Probabilistic Homotopy Optimization was evaluated on cart-pole swing-up and MIT Humanoid flips. On cart-pole, PHO and the RRT-inspired baseline RHO solved many goal instances that linear-interpolation homotopy and direct root solves could not under a budget of 200 solver queries. On humanoid back and front flips, LI-HO solved quickly but with higher objective values, whereas PHO and RHO reached at least 80% success around 700 seconds and PHO continued to improve solution quality over time (Pardis et al., 2024).

Sparse recovery and topology optimization show that the paradigm is not restricted to classical feasibility constraints. In $t$7-regularized least squares, the regularization path

$t$8

is piecewise constant in $t$9, and the Continuation Single Best Replacement and $H(z,\tau)=0.$00 Regularization Path Descent algorithms exploit this path structure to track supports across decreasing $H(z,\tau)=0.$01 (Soussen et al., 2014). In higher-order topology optimization, continuation and primal-dual barrier relaxation maintained strict feasibility of the density field $H(z,\tau)=0.$02 throughout 26 successful homotopy steps on a compliance-minimization problem over a mesh with 11100 triangles and 5711 vertices (Gangl et al., 7 Jan 2026).

A broader, non-optimization use of the idea appears in polymer physics. There the relaxation of topological constraints in ring-linear blends is quantified through the periodic Gauss linking number $H(z,\tau)=0.$03; threading is detected by $H(z,\tau)=0.$04, and in majority-linear blends the ring diffusive relaxation time scales as

$H(z,\tau)=0.$05

This usage is conceptually related because relaxation is again organized by a topological obstruction and its gradual release, although the underlying mechanism is reptation rather than numerical continuation (Vigil et al., 2024).

6. Comparative structure, limitations, and recurring difficulties

A persistent theme is that the success of a homotopy-based relaxation depends less on the existence of a continuation parameter than on how the relaxation encodes multiplier structure, curvature, and feasible-set geometry. In MPSC, the Kanzow–Schwartz relaxation succeeds where Scholtes and Steffensen–Ulbrich may fail because the quadrant-wise NCP construction supports recovery of separate $H(z,\tau)=0.$06- and $H(z,\tau)=0.$07-multipliers satisfying the M-stationarity condition at double-active indices (Kanzow et al., 2018). In MPCCs, the Lasry–Lions envelope is distinguished from plain Moreau-type quadratic penalties by its $H(z,\tau)=0.$08 regularity and explicit Lipschitz gradient, which make fast first-order and quasi-Newton methods more natural (Wang et al., 7 Jul 2025). In conic optimization, self-concordant barriers provide local norms and descent inequalities unavailable in ad hoc smoothings, but feasibility is obtained only in the interior of the cone and high accuracy still requires a growing homotopy parameter (Dvurechensky et al., 2022).

Several limitations recur across the literature. Global minimization is generally not guaranteed in nonconvex settings. Switching-constrained and complementarity-constrained homotopies can converge to local stationary points, and branch structure can induce many local minima (Kanzow et al., 2018). PHO does not guarantee solution of the goal NLP even when a feasible terminal problem exists, because success depends on solver basins, the parameterization $H(z,\tau)=0.$09, and computational budget (Pardis et al., 2024). In topology optimization, ill-conditioning increases as the barrier parameter $H(z,\tau)=0.$10 decreases, and smaller homotopy steps are often required near the terminal design because major topological changes concentrate there (Gangl et al., 7 Jan 2026). In Bayesian relaxation, very large $H(z,\tau)=0.$11 can force HMC step sizes to become very small because the Hessian of the negative log-posterior acquires $H(z,\tau)=0.$12-scaled terms (Duan et al., 2018). In feasible-set convex homotopies, unbounded intermediate sets or nonsmooth boundaries can invalidate the ODE theory (Klingler et al., 2024).

Constraint qualification and identifiability issues remain technically central. MPSC-NNAMCQ, BCCQ, BCQ$H(z,\tau)=0.$13, GCQ, and related tailored CQs appear because standard MFCQ or LICQ are frequently unavailable in the original formulation (Kanzow et al., 2018, Wang et al., 7 Jul 2025, Potschka et al., 2019). Verifying these assumptions is often nontrivial, and several papers recommend numerical diagnostics rather than exact certification. This suggests that, in practice, homotopy-based constraint relaxation is most effective when theoretical regularity, solver design, and parameter scheduling are developed jointly rather than treated as separate concerns.

The comparative record of the cited methods supports a general conclusion: homotopy-based constraint relaxation is a methodology for converting hard constraints into a path of tractable surrogates while preserving meaningful asymptotic information about the original problem. Its strongest results arise when the surrogate family is both analytically faithful to the constraint geometry and numerically compatible with warm-started continuation.