Constrained Optimization on Matrix Lie Groups via Interior-Point Method

Constrained Optimization on Matrix Lie Gr oups via Interior -P oint Method Acl ´ ecio J. Santos 1 , Jean C. Pereira 2 , Guilherme V . Raf fo 1 , 3 Abstract — This paper proposes an interior -point framework for constrained optimization pr oblems whose decision variables evolv e on matrix Lie gr oups. The proposed method, termed the Matrix Lie Gr oup Interior-P oint Method (MLG-IPM), operates directly on the gr oup structure using a minimal Lie algebra parametrization, av oiding redundant matrix r epresentations and eliminating explicit dependence on Riemannian metrics. A primal-dual formulation is developed in which the New- ton system is constructed through sensitivity and curvature matrices. Also, multiplicative updates are performed via the exponential map, ensuring intrinsic feasibility with respect to the group structur e while maintaining strict positivity of slack and dual v ariables thr ough a barrier strategy . A local analysis establishes quadratic conver gence under standard regularity assumptions and characterizes the beha vior under inexact New- ton steps. Statistical comparisons against Riemannian Interior - Point Methods, specifically f or optimization pr oblems defined over the Special Orthogonal Group SO( n ) and Special Linear Group SL( n ) , demonstrate that the proposed approach achie ves higher success rates, fewer iterations, and superior numerical accuracy. Furthermore, its rob ustness under perturbations suggests that this method serves as a consistent and reliable alternativ e for structured manif old optimization. Index T erms— Lie Group, Matrix Lie Group, IPM, Constrained Optimization, Interior-P oint Method. I . I N T RO D U C T I O N Optimization algorithms are widely used in robotics, con- trol, and machine learning, as they enable the computation of optimal solutions for problems in volving multiple criteria and constraints. In robotics and control specifically, decision variables often represent configurations, poses, and spatial transformations of physical systems. In these settings, in addition to minimizing a cost functional, it is necessary to satisfy physical limits, safety requirements, and the underly- ing system dynamics. This ensures that the resulting solution is not only optimal but also operationally feasible [1]. In many of these problems, the variables do not belong to R n , but rather to matrix Lie groups that model geometric transformations, such as rotations and rigid body motions. ©2026. This manuscript version is made available under the CC-BY - NC-ND 4.0 license https://creativ ecommons.org/licenses/by-nc-nd/4.0/. This paper was not presented at any IF AC meeting. This work was supported by the Brazilian agencies Coordination for the Improv ement of Higher Education Personnel (CAPES) – Finance Code 001, CNPq under grants 317058/2023-1 and 422143/2023-5, and F APEMIG. 1 A. J. Santos and G. V . Raffo are with the Graduate Program in Electrical Engineering, Univ ersidade Federal de Minas Gerais, A v . Ant ˆ onio Carlos 6627, 31270-901, Belo Horizonte, MG, Brazil. { aclecio1999, raffo } @ufmg.br 2 J. C. Pereira is with Department of Mechatronics Engineering, Campus Divin ´ opolis — CEFET –MG, Divin ´ opolis, MG, 35503-822, Brazil 3 G. V . Raffo is also with the Department of Electronic Engineering, Univ ersidade Federal de Minas Gerais. This is the case when working with rotation matrices and homogeneous transformations, which must satisfy structural properties including orthogonality , unit determinant, closure, and in vertibility [2]. When these problems are formulated in standard Euclidean spaces, additional nonlinear constraints must be explicitly imposed to ensure the solution remains within the matrix group. Howe ver , this approach increases both the analytical and computational complexities [3]. T o account for this geometric structure, specialized ap- proaches have been dev eloped to perform optimization di- rectly on differentiable manifolds and, in particular, on matrix Lie groups. The most prominent among these are Riemannian optimization methods, which explicitly exploit the group’ s geometry through tools such as the tangent space, Riemannian metrics, and the exponential map. Ac- cordingly , the optimization iterations are designed to remain intrinsically on the group, thereby eliminating the need for additional constraints to preserve algebraic properties [4]. In this context, Riemannian methods can be categorized, analogous to the Euclidean case, into first- and second-order approaches, depending on whether they use only gradient information or incorporate second-order curvature. Among first-order methods, Riemannian gradient descent stands out, where the descent direction is computed within the tangent space, while the update is executed via the exponential map or a retraction to ensure the iterates remain on the group [5], [6], [7]. Second-order methods, such as the Riemannian Newton method, use the Riemannian Hessian, defined as the cov ariant deriv ative of the gradient, to exploit curvature information and derive more efficient search directions [8]. When optimization problems in volv e additional equality and inequality constraints alongside the group structure, extensions that combine Riemannian geometry with classical constrained optimization techniques are required. In this setting, Riemannian interior-point and barrier methods are particularly relev ant. These approaches formulate barrier functions that are compatible with the matrix Lie group structure, with search directions computed within the tangent space. Such methods are critical for control and motion planning problems, where the geometric integrity of the Lie group must be preserved while simultaneously satisfying op- erational constraints in a systematic and numerically robust manner [9], [10], [11]. Despite their conceptual adv antages, optimization methods on matrix Lie groups still present structural and numeri- cal challenges. In particular, the matrix representation of the tangent space may introduce redundancies and linear dependencies associated with the chosen parametrization. Consequently , dif ferent matrices may represent the same geometric direction, adversely affecting numerical condition- ing and computational efficiency [12]. Moreover , existing Riemannian formulations inherently depend on the choice of a Riemannian metric, which determines the explicit form of the gradient and the Hessian. Howe ver , a bi-in v ariant Riemannian metric is not alw ays a vailable for a giv en group [13], [14]. These limitations motiv ate the development of op- timization algorithms that operate with minimal representa- tions of matrix Lie groups, thereby eliminating redundancies and reducing the explicit dependence on metric structures and auxiliary geometric operations. In this work, we introduce the MLG-IPM (Matrix Lie Group Interior -Point Method), an optimizer designed to solve constrained optimization problems whose decision variables belong to general matrix Lie groups. The proposed method operates directly on the group structure and uses a minimal parametrization of the gradient. This approach effecti vely av oids the redundancies inherent in standard matrix represen- tations and eliminates the need to explicitly define a Rieman- nian metric. In addition to the algorithmic framew ork, we provide a local con ver gence analysis, establishing theoretical guarantees for the method’ s beha vior in the neighborhood of an optimal solution. Notation: Scalars and vectors are denoted by lowercase letters, while stacked vectors composed of subvectors are denoted by bold lo wercase letters. Matrices are denoted by bold uppercase letters, and R denotes the set of real numbers. Let G be a matrix Lie group with Lie algebra g , and let S : R m → g be a linear map. V ariations on the group are expressed via the exponential map exp( · ) . W e consider a tu- ple of matrices X ≜ { X 1 , . . . , X n } , with each X i ∈ G . For a stacked vector ζ ≜ [ ζ ⊤ 1 , . . . , ζ ⊤ n ] , where ζ i ∈ R m i , we de- fine X exp ( S ( ζ )) ≜ { X 1 exp( S ( ζ 1 )) , . . . , X n exp( S ( ζ n )) } , which represents perturbations along the Lie algebra of each component. The identity matrix is denoted by I , and diag ( · ) forms a diagonal matrix from its arguments. I I . P R E L I M I N A R I E S This section establishes the mathematical foundations for the proposed framework. W e revie w essential differential constructions on matrix Lie groups and deriv e first- and second-order linearizations via local Lie algebra coordinates. These deri vations provide the sensiti vity and curv ature ma- trices that replace the classical Jacobian and Hessian. Definition 1: ([15]) A matrix Lie group is a subgroup G of GL ( n ; C ) with the follo wing property: if A m is any sequence of matrices in G , and A m con verges to some matrix A , then either A is in G or A is not inv ertible. Definition 2: ([12]) Let G be an m -dimensional Lie group and G ∈ G . Given a linear map S : R m → g and a differentiable function f : G → R , the gradient D f [ G ] is defined by D f [ G ] ζ = lim ε → 0 f  G exp( S ( ζ ) ε )  − f ( G ) ε , ε ∈ R , (1) where ζ ∈ R m denotes the generalized twist, and D f [ G ] =  ℓ 1 ( G ) ℓ 2 ( G ) · · · ℓ m ( G )  , ℓ m : G → R . Remark 1: ([12]) Equiv alently , the gradient D f [ G ] admits the differential representation D f [ G ] ζ = d dε f  G exp  S ( ζ ) ε    ε =0 . Definition 3: Let G = { G 1 , . . . , G n } be a finite col- lection of group elements. The gradient of the scalar function f e valuated on G is defined as ∇ G f =  D f [ G 1 ] D f [ G 2 ] · · · D f [ G n ]  ∈ R 1 × ( n · m ) . Definition 4: Let f = [ f 1 , . . . , f l ] ⊤ : G → R l be a differentiable mapping, where each component f j : G → R is scalar-valued. The Sensitivity Matrix of f at G is defined as the matrix representation of the dif ferential, J G f =    D f 1 [ G 1 ] · · · D f 1 [ G n ] . . . . . . . . . D f l [ G 1 ] · · · D f l [ G n ]    ∈ R l × ( n · m ) . Definition 5: Let f : G → R be twice differentiable. The Curv ature Matrix of f at G is defined as the matrix representation of the second differential, namely H G f = J G ( ∇ G f ) ⊤ , where H G f ∈ R ( n · m ) × ( n · m ) . Remark 2: Although termed Sensitivity Matrix and Cur- vature Matrix, these objects correspond to the first- and second-order linearizations of the mapping in Lie algebra coordinates, playing roles analogous to the Jacobian and Hessian on R n in Ne wton-type methods. I I I . M ATR I X L I E G R O U P I N T E R I O R - P O I N T M E T H O D This section details the proposed interior-point frame work formulated directly on matrix Lie groups. Drawing on the foundations in [9], [16], we extend the classical primal–dual interior-point paradigm to optimization problems where de- cision variables ev olve on smooth matrix manifolds endowed with a Lie group structure. Consider the nonlinear optimization problem min G f ( G ) s.t. g i ( G ) ≤ 0 , i = 1 , . . . , n 1 , h j ( G ) = 0 , j = 1 , . . . , n 2 . (2) The objective function f : G → R and the constraint mappings g i : G → R and h j : G → R are assumed to be twice continuously differentiable. W e further assume that the feasible set is nonempty and contains at least one strictly feasible point such that g i ( G ) ≤ 0 and h j ( G ) = 0 for all i, j . A. Lagrangian and primal–dual formulation The primal-dual formulation provides a more operational perspectiv e through the Lagrangian associated with the prob- lem (2), L ( G , ν, λ ) = f ( G ) + ν ⊤ g ( G ) + λ ⊤ h ( G ) , with ν ∈ R n 1 and λ ∈ R n 2 denoting the Lagrange multipliers associated with the inequality and equality constraints, re- spectiv ely . The primal-dual interior-point framework reformulates inequality constraints as equalities by introducing slack variables s ∈ R n 1 , such that g ( G ) + s = 0 , s i > 0 , i = 1 , · · · , n 1 . This transformation enforces strict fea- sibility through the positivity of s . Consequently , the Karush–Kuhn–T ucker (KKT) conditions are defined by pri- mal feasibility of the augmented equality system, dual feasi- bility , Lagrangian stationary on the Lie group, and comple- mentarity between slack and dual variables. The stationarity condition is obtained by differentiating the Lagrangian with respect to G : ∇ G L = ∇ G f +( J G g ) ⊤ ν + ( J G h ) ⊤ λ. Newton-type primal–dual methods require second-order in- formation, provided by the Lagrangian curvature matrix, H G L = H G f + P n 1 i =1 ν i H G g i + P n 2 j =1 λ j H G h j . If the inequality constraints are linear , their second-order deriv ati ves vanish. In such cases, the Curvature Matrix coincides with that of the objectiv e function augmented only by the second-order terms associated with the equality constraints. Under these considerations, the primal-dual KKT con- ditions take the form ∇ G L = 0 , g ( G ) + s = 0 , h ( G ) = 0 , S ν − µe = 0 , where S = diag( s ) , µ > 0 is the barrier parameter , and e is the vector of ones. The classical complementarity condition s i ν i = 0 is replaced by the relaxed relation s i ν i = µ, which guarantees that all iterates remain strictly in the interior of the feasible region [16]. T o apply Ne wton’ s method, we define the v ector field F =  ( ∇ G L ) ⊤ ( g ( G ) + s ) ⊤ h ( G ) ⊤ ( S ν − µe ) ⊤  ⊤ . The stacked primal–dual variable is defined as z ≜  ν ⊤ λ ⊤ s ⊤  ⊤ ∈ R d , with d ≜ p + 2 m . The primal–dual variables are then embedded into the af fine matrix structure Z ≜  I d z 0 1 × d 1  ∈ T ( d ) , where T ( d ) denotes the d -dimensional translation group, represented by affine matrices whose action corre- sponds to translations in R d [12]. This representation allows multiplicativ e updates through the exponential map while preserving positi vity and the complementarity structure. Linearizing the KKT conditions around the current iterate X k = {G k , Z k } leads to the primal–dual Newton system J X k F · ∆ x k = − F ( X k ) , (3) where the search direction is defined as ∆ x k = [(∆ x k 1 ) ⊤ , · · · , (∆ x k n ) ⊤ , (∆ x n +1 ) ⊤ ] ⊤ ∈ R nm + d , with ∆ x k 1: n ∈ R m , ∆ x k n +1 ∈ R d , and J X k F ev aluated at X being giv en by J X k F =      H G k L ( J G k g ) ⊤ J G k h ) ⊤ 0 J G k g 0 0 I J G k h 0 0 0 0 S k 0 diag ( ν k )      . After solving the Newton system, the primal variables ev olve on the Lie group according to X k +1 = X k exp  S ( α ∆ x k )  , (4) where the update is applied componentwise, with X k = { G k 1 , · · · , G k n , Z k } . The step size α ∈ (0 , 1] is determined using a fraction- to-the-boundary strategy in order to preserve strict pos- itivity of the slack and dual v ariables. T o this end, we compute the maximum admissible primal and dual step lengths as α pri = min i : ∆ s i < 0 n − s i ∆ s i o , α dual = min i : ∆ ν i < 0 n − ν i ∆ ν i o , where the minimum ov er an empty index set is interpreted as + ∞ . The actual step size is Algorithm 1 Matrix Lie Group Interior-Point Method 1: Input: Initial strictly feasible X 0 with s 0 > 0 , ν 0 > 0 ; tolerance ε tol > 0 ; parameters σ ∈ (0 . 1 , 0 . 5) , τ ∈ (0 , 1) . 2: Set k ← 0 . 3: while ∥ F ( X k ) ∥ > ε tol do 4: Compute F k . 5: Form the Newton matrix J X k F . 6: Solve the linear system J X k F ∆ x k = − F k . 7: Compute maximum feasible step sizes: α pri , α dual 8: Set α k = τ min  1 , α pri , α dual  . 9: Update primal Lie group variables: X k +1 = X k exp  S ( α k ∆ x k )  . 10: Recov er X k . 11: Update barrier parameter: µ k +1 = σ ( s k +1 ) ⊤ ν k +1 m . 12: k ← k + 1 . 13: end while 14: Output: X k . then defined by α = τ min  1 , α pri , α dual  , with τ ∈ (0 , 1) ensuring that s k +1 > 0 and ν k +1 > 0 . The complementarity parameter is then updated according to µ = σ s ⊤ ν m , with σ ∈ [0 . 1 , 0 . 5] . The iterations are repeated until µ becomes sufficiently small. The complete MLG-IPM procedure is summarized in Algorithm 1. Remark 3: Multiplicative updates via the exponential map ensure that iterates remain on the Lie group at ev ery step. In contrast to purely additive first-order schemes [17], [18], [19], this approach inherently incorporates higher-order ge- ometric information derived from the group structure. I V . L O C A L C O N V E R G E N C E A NA L Y S I S This section presents the local con ver gence analysis for the proposed Newton-type method on X . Drawing on the framew orks in [9], [16], we suitably adapt classical results to accommodate the primal-dual formulation and the underlying Lie group geometry . All assumptions and conv ergence results are reformulated to be consistent with the geometric notation established throughout this work. Assumption 1: There exists a X ∗ satisfying the KKT conditions, i.e., F ( X ∗ ) = 0 . Assumption 2: The set {∇ G ∗ h j } n 2 j =1 ∪ {∇ G ∗ g i } i ∈ A ( G ∗ ) is linearly independent, where the activ e set is defined as A ( G ∗ ) := { i | g i ( G ∗ ) = 0 } . Assumption 3: ν ∗ i > 0 whenev er g i ( G ∗ ) = 0 , i = 1 , . . . , n 1 . Assumption 4: The Lagrangian curvature matrix satisfies ζ ⊤ H G ∗ L ζ > 0 for ev ery nonzero vector ζ ∈ R n · m such that ∇ G ∗ h j ζ = 0 , j = 1 , . . . , n 2 , and ∇ G ∗ g i ζ = 0 , i ∈ A ( G ∗ ) . Theor em 1: ([9]) If Assumptions 1 – 4 hold at some point G ∗ , then the Sensitivity Matrix J X ∗ F is nonsingular . Theor em 2: Under Assumptions 1-4, there exists a neigh- borhood U of X ∗ such that, for any initial point X 0 ∈ U sufficiently close to X ∗ , the iteration J X k F ∆ x k = − F ( X k ) + ρ k , (5) combined with the exponential update (4), is well defined and generates a sequence {X k } conv erging to X ∗ . The local con vergence rate is go verned by the step size α and residual ρ k as follo ws: For α =1 : (i) Quadratic con vergence if ρ k =0 or ∥ ρ k ∥ = O ( ∥ F ( X k ) ∥ 2 ) ; (ii) Linear con vergence if ∥ ρ k ∥ = O ( ∥ F ( X k ) ∥ ) . For 0 <α< 1 : (i) Linear con vergence if ρ k =0 or ∥ ρ k ∥ = O ( ∥ F ( X k ) ∥ ) . Pr oof: Applying (1) to the vector field F yields F  X k exp ( S ( ζ ) ε )  = F ( X k ) + J X F ζ ε + O ( ∥ ζ ε ∥ 2 ) , (6) where ζ = [ ζ ⊤ 1 , · · · , ζ ⊤ n , ζ ⊤ n +1 ] ⊤ ∈ R nm + d , with ζ 1: n ∈ R m and ζ n +1 ∈ R d . By setting ζ ε = α ∆ x k and noting that X k +1 = X k exp( S ( α ∆ x k )) , (6) reduces to F ( X k +1 )= F ( X k )+ α J X k F ∆ x k + O ( ∥ α ∆ x k ∥ 2 ) . (7) Substituting the Newton direction from (5) into (7), we obtain F ( X k +1 ) = (1 − α ) F ( X k )+ αρ k + O ( ∥ α ∆ x k ∥ 2 ) . (8) From Theorem 1, the nonsingularity of J X F in the neighbor- hood of X ∗ ensures that ( J X F ) − 1 is bounded. This leads to the estimate ∥ F ( X k +1 ) ∥≤ C 1 ∥ ρ k ∥ +(1 − α ) ∥ F ( X k ) ∥ + C 2 ∥ F ( X k ) ∥ 2 , (9) for some constants C 1 , C 2 > 0 . The con ver gence beha vior is determined by the parameters α and ρ k • Case α = 1 : The linear term vanishes and (9) reduces to ∥ F ( X k +1 ) ∥ ≤ C 1 ∥ ρ k ∥ + C 2 ∥ F ( X k ) ∥ 2 . In this case, the behavior of the method is mainly governed by the residual ρ k . Quadratic conv ergence is achie ved if ∥ ρ k ∥ = O ( ∥ F ( X k ) ∥ 2 ) (including ρ k = 0 ), whereas ∥ ρ k ∥ = O ( ∥ F ( X k ) ∥ ) results in linear conv ergence. • Case 0 < α < 1 : The term (1 − α ) ∥ F ( X k ) ∥ introduces a linear component in the estimate, which dominates the asymptotic behavior unless the residual ρ k decays sufficiently fast. In particular, the method exhibits linear con vergence when ∥ ρ k ∥ = O ( ∥ F ( X k ) ∥ ) or ∥ ρ k ∥ = 0 . Finally , the nonsingularity of J X F ( X ∗ ) ensures that ∥ F ( X k ) ∥ → 0 that implies X k → X ∗ , completing the proof. V . C O M P A R A T I V E S T U DY This section ev aluates the performance of the proposed MLG-IPM through a quantitati ve comparison with the Rie- mannian Interior-Point Method (RIPM) [9] and a qualita- tiv e assessment against a Euclidean Interior-Point Method (EIPM) [16]. The quantitative analysis inv olved 1000 in- dependent simulations on SO (7) and SL (7) per method, measuring success rate, computational time, number of iterations, and residual error . Because the Lilliefors test ( γ = 0 . 05 ) rejected normality for the continuous metrics, we employed the nonparametric W ilcoxon–Mann–Whitney test for comparisons. Categorical outcomes were analyzed via the chi-square test and the two-proportion Z-test for the null hypothesis H 0 : p MLG = p RIPM , where p MLG and p RIPM denote the success probabilities of MLG-IPM and RIPM, respectiv ely . Finally , ef fect sizes were quantified using Cram ´ er’ s V for categorical variables and Cohen’ s d for continuous metrics. A. Comparison between RIPM and MLG-IPM in SO (7) The optimization problem considered is min G 1 , G 2 ∈ SO (7) 1 2 2 X i =1 ∥ log ( G d ⊤ G i ) ∥ F s.t. G i (1 , 1) − 0 . 5 ≤ 0 , G i (2 , 2) − 0 . 3 ≤ 0 , (10) with G 1 , G 2 , and G d randomly sampled in SO (7) at each run. T ABLE I S T A T IS T I C AL C OM PA RI S O N B ET W E E N M L G - IP M A N D R I PM W IT H ρ k = 0 I N SO (7) . Metric MLG-IPM RIPM T ime (s) Median 10.5661 20.8154 T ime (s) Mean 13 . 6649 ± 25 . 2225 18 . 4179 ± 7 . 4139 Iterations Median 20.00 500.00 Iterations Mean 25 . 76 ± 47 . 84 394 . 31 ± 139 . 98 Success Rate 0.991 0.416 Error Mean 5 . 0 × 10 − 5 ± 4 . 17 × 10 − 4 4 . 7 × 10 − 3 ± 1 . 7 × 10 − 2 1) W ithout P erturbation: T able I summarizes the perfor- mance of both methods for ρ k = 0 , revealing not only statistical differences but also meaningful practical impli- cations. The success rates highlight a fundamental gap in robustness: MLG-IPM conv erges in nearly all executions, whereas RIPM fails in a significant portion of cases. The strong association between optimizer choice and con ver gence (Cram ´ er’ s V = 0 . 628 ) confirms that the probability of success is highly dependent on the selected method. Computationally , MLG-IPM achiev es solutions faster and with fewer iterations. Conv ersely , the high frequency with which RIPM reaches the iteration limit suggests stagnation near the stopping threshold rather than stable con vergence. This indicates a structural difference in behavior: while MLG-IPM progresses steadily tow ard optimality , RIPM of- ten fails to satisfy termination criteria within reasonable bounds. Regarding accurac y , MLG-IPM yields a lower mean final error across successful runs, indicating con ver gence closer to the optimal solution. The larger errors and greater dispersion observed for RIPM reflect reduced numerical precision. Col- lectiv ely , these results establish the superiority of MLG-IPM across three dimensions: reliability (success rate), efficienc y (computational cost), and accuracy (final residual). F or ρ k = 0 , the MLG-IPM adv antages are both statistically significant and structurally decisive. 2) W ith P erturbation: T able II reports the results obtained with residual based perturbation, ρ k = 0 . 01 ∥ F k ∥ . Although the regularization slightly modifies the numerical behavior of both methods, the overall performance hierarchy remains unchanged. T ABLE II S T A T IS T I C AL C OM PA RI S O N B ET W E E N M L G - IP M A N D R I PM W IT H ρ k = 0 . 01 ∥ F k ∥ I N SO (7) . Metric MLG-IPM RIPM T ime (s) Median 11.44 22.29 T ime (s) Mean 18 . 88 ± 40 . 89 18 . 94 ± 7 . 50 Iterations Median 21.00 500.00 Iterations Mean 35 . 12 ± 77 . 52 393 . 45 ± 141 . 44 Success Rate 0.973 0.388 Error Mean 4 . 940 × 10 − 3 ± 1 . 0 × 10 − 6 9 . 9 × 10 − 3 ± 1 . 7 × 10 − 2 The success rate continues to fav or MLG-IPM by a wide margin, indicating that the perturbation does not significantly alter its robustness. The chi-square and Z- tests confirm statistical significance ( p < 0 . 001 ), and the large effect size (Cram ´ er’ s V = 0 . 626 ) reinforces that optimizer choice remains strongly associated with con ver gence outcome ev en under regularization. In terms of computational effort, the Wilcoxon-Mann- Whitney tests again reject H 0 for both time and iterations. As shown in T able II, MLG-IPM maintains lower median time and substantially fewer iterations. Although its mean time presents higher dispersion, this behavior stems from occasional longer runs rather than systematic inef ficiency . In contrast, RIPM continues to accumulate results at the maxi- mum iteration limit, indicating persistent structural dif ficulty in meeting stopping criteria despite the perturbation. Regarding solution quality , the final error values confirm that MLG-IPM preserv es superior numerical accurac y among successful runs. The W ilcoxon-Mann-Whitney test detects statistical significance, and the dif ference in mean error sug- gests that residual based regularization does not compensate for the intrinsic conv ergence limitations observed in RIPM. B. Comparison between RIPM and MLG-IPM in SL (7) For the comparison in SL (7) , we consider the following optimization problem: min G 1 , G 2 ∈ SL (7) 1 2 2 X i =1 T r ( G d − 1 G i ) 2 s.t. − Tr ( G d − 1 G i ) 2 + 0 . 2 ≤ 0 , (11) with G 1 , G 2 , and G d randomly sampled in SL (7) at each run. T ABLE III S T A T IS T I C AL C OM PA RI S O N B ET W E E N M L G - IP M A N D R I PM W IT H ρ k = 0 I N SL (7) . Metric MLG-IPM RIPM T ime (s) Median 0.1187 6.6486 T ime (s) Mean 0 . 2590 ± 0 . 4097 7 . 60 ± 7 . 83 Iterations Median 58 500 Iterations Mean 66 . 98 ± 63 . 31 369 ± 188 . 1 Success Rate 0.981 0.359 Error Mean 6 . 6 × 10 − 5 ± 2 . 0 × 10 − 3 2 . 7 × 10 − 3 ± 1 . 4 × 10 − 2 1) W ithout P erturbation: T able III shows that MLG-IPM con verged in 98.1% of ex ecutions, whereas RIPM succeeded in only 35.9% (an absolute difference of 62.2 percentage points). The high Cram ´ er’ s V ( V = 0 . 660 ) indicates a strong association between optimizer choice and con ver gence probability , highlighting a substantial robustness adv antage for MLG-IPM. T o formally assess the difference in success proportions, we performed a chi-square test, yielding χ 2 = 872 . 1 . Although the reported p value was p = 1 , the Z-test for proportions resulted in Z = 29 . 6 with p < 0 . 05 , indicating a statistically significant difference. The extremely large Z statistic reinforces that the observed discrepancy is unlikely due to random v ariation, consistent with the lar ge effect size measured by Cram ´ er’ s V . Computationally , both time and iteration distributions were identified as non normal (Lilliefors test, p < 0 . 05 ), so we conducted comparisons using the W ilcoxon-Mann-Whitney test. The differences were statistically significant ( p < 0 . 05 ) for both metrics. The median ex ecution time of MLG-IPM (0.1187 s) is much smaller than that of RIPM (6.6486 s), yielding a median dif ference of approximately 6.53 seconds. Similarly , the median iteration count (58 vs. 500) indicates that RIPM frequently reaches the maximum iteration limit, suggesting stagnation rather than natural conv ergence. Regarding solution accuracy , considering only successful runs, MLG-IPM achiev ed a substantially smaller mean final error ( 6 . 6 × 10 − 5 ) compared to RIPM ( 2 . 68 × 10 − 3 ). The W ilcoxon-Mann-Whitney test confirmed statistical signifi- cance ( p < 0 . 05 ). Moreov er , RIPM exhibits greater error dispersion, indicating reduced numerical consistency . T ABLE IV S T A T IS T I C AL C OM PA RI S O N B ET W E E N M L G - IP M A N D R I PM W IT H ρ k = 0 . 001 ∥ F k ∥ I N SL (7) . Metric MLG-IPM RIPM T ime (s) Median 0.1346 6.8945 T ime (s) Mean 0 . 36 ± 0 . 60 6 . 28 ± 6 . 77 Iterations Median 62 500 Iterations Mean 106 . 15 ± 123 . 96 344 . 54 ± 197 . 3 Success Rate 0.937 0.408 Error Mean 5 × 10 − 4 ± 5 × 10 − 3 2 × 10 − 3 ± 1 × 10 − 2 2) W ith P erturbation: T able IV presents the statistical comparison between MLG-IPM and RIPM based on 1000 independent runs per method for problem (11). The results rev eal statistically significant and practically meaningful dif- ferences across all ev aluated metrics. In terms of reliability , MLG-IPM con ver ged in 93.7% of ex ecutions, whereas RIPM succeeded in only 40.8% . The chi-square test ga ve χ 2 = 632 . 90 , and although the reported p v alue was p = 1 . 00 , the Z-test for proportions yielded Z = 25 . 21 with p < 0 . 05 , indicating a statistically significant difference. Furthermore, Cram ´ er’ s V = 0 . 563 indicates a large effect size, confirming a strong association between optimizer choice and conv ergence probability . Computationally , the Lilliefors test indicated that both time and iteration distributions deviate from normality ( p < 0 . 05 ), so we used the W ilcoxon-Mann-Whitney test. The difference in ex ecution time was statistically significant ( p < 0 . 05 ), with MLG-IPM presenting a substantially lower median time (0.1346 s) compared to RIPM (6.8945 s). A similar pattern emerged for iteration count. MLG- IPM required significantly fewer iterations (median of 62) than RIPM (median of 500), which frequently reached the maximum iteration limit, suggesting stagnation rather than natural con vergence. The W ilcoxon test confirmed statistical significance ( p < 0 . 05 ), with a median difference of 438 iterations. Regarding numerical accuracy , considering only successful runs, MLG-IPM achieved a lower mean final error ( 5 . 22 × 10 − 4 ) compared to RIPM ( 1 . 67 × 10 − 3 ). This difference was also statistically significant according to the W ilcoxon- Mann-Whitney test ( p < 0 . 05 ). Additionally , RIPM e xhibited greater error dispersion, indicating lo wer numerical consis- tency . C. Comparison between EIPM and MLG-IPM W e compare the EIPM [16] and the proposed MLG- IPM through a structural and algorithmic analysis. The comparison focuses on three main aspects: the underlying geometric framework, the treatment of structural constraints, and the computational implications of solving the Ne wton- type systems arising at each iteration. a) Geometric setting: EIPM is formulated in Euclidean space, where structural constraints are imposed explicitly through equality conditions. In contrast, MLG-IPM operates directly on a matrix Lie group, treating the feasible set as a smooth manifold endowed with group structure. b) T r eatment of constraints: In EIPM, properties such as orthogonality and determinant conditions are enforced via equality constraints and Lagrange multipliers. In MLG- IPM, these properties are intrinsic to the Lie group and are therefore incorporated directly into the search space. This leads to a more compact problem formulation. c) Computational dimension: Both methods require the solution of a Newton-type linear system at each iteration. In EIPM, the system dimension includes primal variables and multipliers associated with equality constraints. In MLG- IPM, the search direction is computed in the Lie algebra, whose dimension equals that of the group. When a group structure replaces explicit constraints, this may reduce the dimension of the linear system and potentially improve computational ef ficiency . d) Modeling capability: Problems defined over struc- tured matrix sets can be dif ficult to treat in EIPM due to nonlinear equality constraints. MLG-IPM handles such prob- lems naturally by preserving the group structure throughout the iterations. V I . C O N C L U S I O N This work introduced MLG-IPM as an interior-point method formulated directly on matrix Lie groups through a minimal Lie algebra parametrization, providing a geometri- cally consistent and compact framework for constrained opti- mization on non-Euclidean manifolds. By computing search directions in local exponential coordinates and performing multiplicativ e updates that preserve the intrinsic structure of the group, the method avoids redundant parametrizations and explicit metric constructions while maintaining the main the- oretical properties of primal–dual interior -point schemes. The local con ver gence analysis established quadratic conv ergence under standard assumptions and characterized the influence of inexact Newton steps on the conv ergence rate. Numerical experiments indicated that MLG-IPM achiev es higher suc- cess rates, fewer iterations, and impro ved numerical accurac y when compared to RIPM, while maintaining robustness under perturbations. These results suggest that MLG-IPM is a practical approach for constrained optimization on matrix Lie groups, with potential applications in robotics, geometric control, and estimation on nonlinear manifolds. 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