Open research questions in Advanced Optimization Algorithms Research

What the literature leaves open

• Further research can be done to extend the results to non-linear vector optimization problems. The concept of e-properly efficient solution can be used to analyze other types of optimization problems.

  Approximate Proper Efficiency in Vector Optimization via Benson’s Approach · 2026 · DOI

• The linear complementarity problem is a widely studied problem, but there is a need for a novel method to solve the sparse linear complementarity problem. The existing methods do not provide a sparse solution to the linear complementarity problem.

  Accelerated nonnegative proximal gradient algorithm for sparse linear complementarity problem · 2026 · DOI

• 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

• Future research can build upon the paper's findings by exploring the applications of geometric analysis in nonlinear optimization. Further research can also investigate the development of numerical algorithms based on the geometric approaches presented in the paper.

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

• In this paper, we focus on local search algorithms for IQP. We propose 4 novel operators and a two-mode algorithm with new scoring functions to enhance the search process. Experiments show that the performance of our solver is outstanding, setting 6 new records for open instances. Future work aims to combine our solver with other solvers to prune the search space and speed up the search process.

  Local Search for Integer Quadratic Programming · 2026 · DOI

• The paper suggests that future research should focus on developing more efficient algorithms for nonlinear optimization. The paper notes that the results can be applied to a wide range of optimization problems, including those with non-relaxable lower-level constraints.

  On constraint qualifications for non-relaxable sets and an augmented Lagrangian method · 2026 · DOI

• 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

• This is due to the fact that a stability analysis regarding the algorithm parameters ρ and step size h remains unexplored, preventing optimal tuning. While we discretized the scheme using numerical methods and confirmed its convergence through experiments, a rigorous discrete-level analysis remains an open problem.

  Convergence Analysis via ODE Approach for Convex Optimization with Linear Equality Constraints · 2026

• A batch of test problems are randomly generated by using the code3 in and the problem information is: Q is sparse with the density n2/4, whose elements are normally distributed, the elements in g are uniform random values in the interval [−10, 10], the bounds l0, u0 are uniform random values in the interval [−10, 10] and.

  A Lasry–Lions Envelope Approach for Mathematical Programs with Complementarity 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 lack of a non-asymptotic convergence guarantee for existing methods. The difficulty in assessing the accuracy of approximations for copositive cones. The need for a method that can solve general copositive programming problems efficiently.

  Inexact subgradient algorithm with a non-asymptotic convergence guarantee for copositive programming problems · 2026 · DOI

• The lack of a strengthened Moment-SOS hierarchy for solving matrix polynomial optimization. The slow convergence speed of the SIP method.

  Lagrange Multiplier Expressions for Matrix Polynomial Optimization and Tight Relaxations · 2026 · DOI

• There is no universally accepted best technique for solving optimization problems, regardless of their formulation. The existing research in nonlinear optimization lacks a broad and flexible convergence theory for inexact restoration applications.

  A general merit function-based global convergent framework for nonlinear optimization · 2026 · DOI

• The paper provides a new result that has no analogue in the literature.

  Approximate Proper Efficiency in Vector Optimization via Benson’s Approach · 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

• Studies have been conducted on copositive cones determined by symmetric cones beyond nonnegative orthants, and on their associated copositive programming problems. We can utilize the mixed-integer linear programming reformulations proposed by Gondzio and Yıldırım to solve the standard quadratic programming problem in (4.7) exactly.

  Inexact subgradient algorithm with a non-asymptotic convergence guarantee for copositive programming problems · 2026 · DOI

• Deriving explicit certificate forms that do not rely on sums of squares. Extending the classical theorems to the matrix context. Constructing a hierarchy of semidefinite optimization relaxations that remain computationally feasible for problems with large numbers of variables and low-dimensional matrix constraints.

  Non-SOS Positivstellensätze for Semialgebraic Sets Defined by Polynomial Matrix Inequalities · 2026 · DOI

• The paper seeks to provide a full characterization of the set of all optimal smoothings for convex cones and sublinear functions.

  The Optimal Smoothings of Sublinear Functions and Convex Cones · 2026 · DOI

• The paper suggests applying the introduced framework to alternatives using line search and trust regions. The paper suggests extending the analysis to other types of methods, including those that do not use iterations with two phases.

  A general merit function-based global convergent framework for nonlinear optimization · 2026 · DOI

• The results rely on certain conditions and assumptions, such as the correlative sparsity pattern and the Archimedean condition - The paper does not provide a comprehensive comparison with existing methods - The scalability and computational tractability of the SDP relaxations may still be limited for very large…

  Non-SOS Positivstellensätze for Semialgebraic Sets Defined by Polynomial Matrix Inequalities · 2026 · DOI

• To further improve the efficiency of the strengthened Moment-SOS hierarchy. To apply the hierarchy to other optimization problems.

  Lagrange Multiplier Expressions for Matrix Polynomial Optimization and Tight Relaxations · 2026 · DOI

Most-cited papers in Advanced Optimization Algorithms Research

• On Nonlinear Fractional Programming · Management Science · 1967 · 2,778 citations
• Technical Note—Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming · Operations Research · 1973 · 1,693 citations
• Minimizing a Submodular Function on a Lattice · Operations Research · 1978 · 925 citations
• Technical Note—A General Inner Approximation Algorithm for Nonconvex Mathematical Programs · Operations Research · 1978 · 903 citations
• Solving Large-Scale Zero-One Linear Programming Problems · Operations Research · 1983 · 486 citations
• Mathematical Programs with Optimization Problems in the Constraints · Operations Research · 1973 · 445 citations
• Non-Linear Programming Via Penalty Functions · Management Science · 1967 · 423 citations
• Inverse Optimization · Operations Research · 2001 · 395 citations
• Feature Article—The Ellipsoid Method: A Survey · Operations Research · 1981 · 358 citations
• Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions · Operations Research · 2009 · 310 citations

Most recent work

• A modified Newmark/Newton–Raphson method with automatic differentiation for general nonlinear dynamics analysis · Nonlinear Dynamics · 2026
• The Lie algebra of XY-mixer topologies and warm starting QAOA for constrained optimization · npj Quantum Information · 2026
• On the optimal parameter values of the Nelder–Mead simplex algorithm · Journal of Computational and Applied Mathematics · 2026
• Implicit differentiation with second-order derivatives and benchmarks in finite-element-based differentiable physics · Computer Physics Communications · 2026
• The Internal Model Principle of Time-Varying Optimization · IEEE Transactions on Automatic Control · 2026
• EDECT algorithm: The hybridization of the filled function and DIRECT methods for global optimization · Numerical Algorithms · 2026
• 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
• Smoothing ℓ1-exact penalty method for intrinsically constrained Riemannian optimization problems · IMA Journal of Numerical Analysis · 2026
• On Coupling Constraints in Pessimistic Linear Bilevel Optimization · Journal of Optimization Theory and Applications · 2026

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