Open research questions in Advanced Optimization Algorithms Research

What the literature leaves open

• Future research could focus on establishing a theoretical guarantee for the existence of a central path when the cone K is nonsymmetric and P = 0. Future research could also explore the application of Clarabel to other types of convex optimization problems.

  Clarabel: An interior-point solver for conic programs with quadratic objectives · 2026 · DOI

• There is a need for a general-purpose interior-point solver for convex optimization problems with conic constraints and quadratic objectives. Existing solvers may not be efficient or effective for these types of problems.

  Clarabel: An interior-point solver for conic programs with quadratic objectives · 2026 · DOI

• The paper identifies a gap in the existing literature on the Frank-Wolfe algorithm, which has primarily focused on convex functions. The paper aims to extend the Frank-Wolfe algorithm to non-convex functions using the concept of star-convexity.

  Frank–wolfe algorithm for star-convex functions · 2026 · DOI

• Linear optimization problems with entropic constraints are challenging due to the entropic constraints. Existing methods have limitations, such as low efficiency or lack of theoretical justification.

  Bregman proximal gradient method for linear optimization under entropic constraints · 2026 · DOI

• The existing algorithms for nonlinear optimization with general equality constraints have limitations. There is a need for a scalable sequential adaptive cubic regularization algorithm.

  A scalable sequential adaptive cubic regularization algorithm for optimization with general equality constraints · 2026 · DOI

• The graph is dynamic and the shortest path needs to be updated frequently. The graph is very large and the shortest path needs to be computed efficiently.

  Incremental Shortest Paths in Almost Linear Time via a Modified Interior Point Method · 2026 · DOI

• The uniform convexity of the problem. The lack of an explicit expression for the solution for p > 2. The need to solve a univariate nonlinear equation of order p.

  Complexity of Minimizing Regularized Convex Quadratic Functions · 2026 · DOI

• The nonconvexity inherent in the LQR problem. The lack of a self-contained and explicit proof of strong duality in the literature. The need for a broader characterization of gradient dominance.

  Revisiting Strong Duality, Hidden Convexity, and Gradient Dominance in the Linear Quadratic Regulator · 2026 · DOI

• The algorithm's convergence rates were previously unknown. The analysis does not generalize to nonsmooth or partially nonsmooth functions. The algorithm's computational performance can be improved.

  Flexible Block-Iterative Analysis for the Frank-Wolfe Algorithm · 2026 · DOI

• The paper identifies the challenge of developing a comprehensive analysis of perturbed optimization. The paper notes the difficulty of establishing sharp bounds and expansions for the difference between the solution of the original problem and its perturbed counterpart. The paper recognizes the need for a novel and unified approach to perturbed optimization, using techniques from convex analysis and smooth optimization.

  Sharp Bounds in Perturbed Smooth Optimization · 2026 · DOI

• Future research can focus on extending the augmented Lagrangian method to handle more complex optimization problems. Future research can also focus on improving the computational efficiency of the method. Future research can explore the application of the method to various fields, including engineering and industrial computation.

  The Augmented Lagrangian Methods: Overview and Recent Advances · 2026 · DOI

• The paper identifies a gap in the existing literature on the augmented Lagrangian method, particularly in its ability to handle nonconvex constraints. The paper aims to fill this gap by providing a unified perspective on constructing augmented Lagrangian functions and discussing recent advancements.

  The Augmented Lagrangian Methods: Overview and Recent Advances · 2026 · DOI

• The paper identifies the challenge of solving indefinite augmented KKT systems on GPUs. The paper notes the challenge of limited parallelism in classical mathematical programming applications. The paper identifies the challenge of increased ill-conditioning in the condensed KKT system.

  Condensed interior-point methods for scalable nonlinear programming on GPUs · 2026 · DOI

• The paper suggests that future research should focus on improving the scalability and efficiency of GPU-accelerated solution strategies. The paper notes that further research is needed to understand the numerical properties of condensed-space IPM strategies.

  Condensed interior-point methods for scalable nonlinear programming on GPUs · 2026 · DOI

• The problem of defining the dissimilarity measure is nonconvex. The algorithm need not be a generator of P, thus the strategy is a mere heuristic for computing the angle Θ(P, Q).

  Measuring Dissimilarity between Convex Cones by Means of Max-Min Angles · 2026 · DOI

• To further develop the proposed mathematical and algorithmic framework. To apply the approach to other classification tasks. To explore the use of other algorithms for computing the dissimilarity measure.

  Measuring Dissimilarity between Convex Cones by Means of Max-Min Angles · 2026 · DOI

• The lack of an accessible iteration bound for the rPDHG algorithm. The need for a two-stage performance analysis of rPDHG. The need for an evaluation of the sensitivity of rPDHG to perturbations in the objective vector.

  Accessible Complexity Bounds for Restarted PDHG on Linear Programs with a Unique Optimizer · 2026 · DOI

• The nonconvexity and nonsmoothness of the problem. The need to leverage the composite structure and the retraction and first-order information of the manifold. The requirement for an efficient method with established oracle complexity and convergence properties.

  An inexact variable metric proximal linearization method for composite optimization on manifolds · 2026 · DOI

• Further research on the application of the proposed method to various fields. Investigation of the extension of the proposed method to more general classes of problems. Development of new methods for solving nonconvex and nonsmooth optimization problems.

  An inexact variable metric proximal linearization method for composite optimization on manifolds · 2026 · DOI

• To further investigate the properties of weighted GMRES. To explore the applications of the paper's results in practice. To develop more efficient preconditioning strategies based on the paper's results.

  Any Nonincreasing Convergence Curves are Simultaneously Possible for GMRES and Weighted GMRES, As Well As for Left and Right Preconditioned GMRES · 2026 · DOI

• The convergence behavior of GMRES is not fully understood. Classical quantities such as the spectrum, pseudo-spectrum, or numerical range do not offer a complete explanation of the observed residual decay.

  Any Nonincreasing Convergence Curves are Simultaneously Possible for GMRES and Weighted GMRES, As Well As for Left and Right Preconditioned GMRES · 2026 · DOI

• The convergence analysis in Theorem 7 assumes the level set L(f(x_k₀)) is bounded, but does not address how Algorithm 1 behaves on unbounded level sets or provide conditions on f that guarantee bounded level sets when combined with high-order Hölderian continuity assumptions.

  Generalized Metric Subregularity with Applications to High-Order Regularized Newton Methods · 2026 · DOI

• The paper proves that under generalized metric subregularity (equation 74), the algorithm achieves convergence rate defined by τ(t), but does not provide concrete characterizations of the admissible function ψ for standard nonconvex problem classes (e.g., sums-of-squares polynomials, neural network training objectives) where metric subregularity can be verified.

  Generalized Metric Subregularity with Applications to High-Order Regularized Newton Methods · 2026 · DOI

• The global convergence analysis of CG methods remains a challenging issue. Large-scale symmetric nonlinear systems of equations are challenging to solve due to expensive Jacobian evaluations and storage.

  Globally convergent RMIL-Type conjugate gradient methods with optimal parameter strategies for large-scale symmetric nonlinear equations · 2026 · DOI

• The paper identifies the challenge of characterizing optimal solutions in nonlinear optimization problems. The challenge of developing numerical algorithms for solving complex optimization problems is also addressed. The paper also discusses the challenge of providing a geometric interpretation of saddle points, convexity in duality theory, and the duality gap as a solution quality measure.

  Teori Dualitas Lagrange dalam Optimasi Non-Linier: Tinjauan Naratif dari Perspektif Analisis Geometris Modern · 2026 · DOI

Most-cited papers in Advanced Optimization Algorithms Research

• Fixed Point Strategies in Data Science · IEEE Transactions on Signal Processing · 2021 · 78 citations
• The Benders Dual Decomposition Method · Operations Research · 2020 · 71 citations
• Inverse Optimization: Theory and Applications · Operations Research · 2023 · 59 citations
• A Classifier to Decide on the Linearization of Mixed-Integer Quadratic Problems in CPLEX · Operations Research · 2022 · 37 citations
• Nonconvex Piecewise Linear Functions: Advanced Formulations and Simple Modeling Tools · Operations Research · 2022 · 34 citations
• Minimizing of the quadratic functional on Hopfield networks · Electronic journal of qualitative theory of differential equations · 2021 · 10 citations
• Optimal Convergence Rates for Goal-Oriented FEM with Quadratic Goal Functional · Computational Methods in Applied Mathematics · 2020 · 9 citations
• A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints · Computational Methods in Applied Mathematics · 2023 · 6 citations
• Instance-Optimal Goal-Oriented Adaptivity · Computational Methods in Applied Mathematics · 2020 · 6 citations
• A 𝑃₁ Finite Element Method for a Distributed Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints · Computational Methods in Applied Mathematics · 2021 · 4 citations

Most recent work

• Accelerated nonnegative proximal gradient algorithm for sparse linear complementarity problem · Computational Optimization and Applications · 2026
• Clarabel: An interior-point solver for conic programs with quadratic objectives · Mathematical Programming Computation · 2026
• An efficient image space algorithm for solving a class of nonconvex programming problems · Journal of Applied Mathematics and Computing · 2026
• Exact Penalty Functions and Global Saddle Points of Augmented Lagrangians for Well-Posed Constrained Optimization Problems · Vietnam Journal of Mathematics · 2026
• Generalized Metric Subregularity with Applications to High-Order Regularized Newton Methods · Mathematics of Operations Research · 2026
• Approximate Proper Efficiency in Vector Optimization via Benson’s Approach · Vietnam Journal of Mathematics · 2026
• ReMU: regional minimal updating for model-based derivative-free optimization · Optimization Methods and Software · 2026
• Globally convergent RMIL-Type conjugate gradient methods with optimal parameter strategies for large-scale symmetric nonlinear equations · Japan Journal of Industrial and Applied Mathematics · 2026
• Teori Dualitas Lagrange dalam Optimasi Non-Linier: Tinjauan Naratif dari Perspektif Analisis Geometris Modern · Griya Journal of Mathematics Education and Application · 2026
• Local Search for Integer Quadratic Programming · INFORMS Journal on Computing · 2026

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