Open research questions in Optimization and Variational Analysis
140 gap statements mined from Optimization and Variational Analysis papers in our 4.5M-paper local library, which holds 1,759 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
Choosing where to publish on Optimization and Variational Analysis? See the ranked Computer Science journals.
What the literature leaves open
One challenge is that the single-valued operator may not be Lipschitz continuous or co-coercive. Another challenge is that the algorithm needs to operate without assuming coercivity or Lipschitz continuity of the single-valued operator. A third challenge is to demonstrate the effectiveness of the algorithm in practice, particularly in signal reconstruction using compressed sensing.
Non-lipschitz inertial contraction-type method for monotone variational inclusion problems · 2026 · DOIThe algorithm's performance is evaluated using numerical experiments, but no theoretical analysis of the algorithm's complexity is provided. The algorithm is compared to existing methods, but the comparison is limited to a few specific methods. The numerical experiments are conducted using a specific set of parameters, which may not be optimal for all problems.
Non-lipschitz inertial contraction-type method for monotone variational inclusion problems · 2026 · DOILack of a comprehensive framework for addressing interval-valued optimization problems on Riemannian manifolds. Insufficient characterization of weak saddle points for the interval-valued Lagrangian function. Limited understanding of the significance of the Riemannian duality framework in addressing non-convex optimization problems.
Wolfe duality and saddle point analysis on Riemannian manifolds for interval-valued optimization problems · 2026 · DOIThe lack of convergence results for a broader class of algorithms for unconstrained optimization. The need for a careful choice of the regularized model at every iteration to achieve full convergence. The importance of employing appropriate matrices to ensure that the sequence of points is variable metric quasi-Fejér monotone.
Prior work has not provided a unified framework for discrete and continuous-time perspectives. There is a need for a novel framework that can achieve linear convergence rates for strongly convex-strongly concave saddle point problems with bilinear coupling.
Nesterov acceleration for strongly convex-strongly concave bilinear saddle point problems: Discrete and continuous-time approaches · 2026 · DOITo the best of our knowledge, constraint qualifications for (NMSIPSC) have not yet been studied in the Euclidean space setting.
On constraint qualifications and optimality conditions for nonsmooth multiobjective semi-infinite programming problems with switching constraints · 2024 · DOIOne challenge is ensuring strong duality in infinite convex optimization. Another challenge is addressing the failure of classical duality schemes to guarantee zero duality gap. The paper also addresses the challenge of deriving general optimality conditions for the dual problems.
Example 4.1 is presented but the detailed analysis and verification that Theorems 4.1 and 4.2 yield non-trivial multiplier rules for this specific infinite programming problem is incomplete. The computational feasibility of constructing the dual problem (D) or (Dₘ) and verifying the multiplier conditions for this example is not demonstrated.
Future research can focus on applying the algorithm to real-world problems. Future research can focus on extending the algorithm to other classes of variational inequality problems. Future research can focus on comparing the algorithm with other existing methods.
A Strongly Convergent Theorem for a Class of Quasimonotone Bilevel Split Variational Inequality Problems · 2026 · DOIThere is a gap in the existing literature because most algorithms require pseudomonotonicity, monotonicity, or inverse strong monotonicity. The existing literature does not provide a comprehensive solution for bilevel variational inequality problems with quasimonotone operators.
A Strongly Convergent Theorem for a Class of Quasimonotone Bilevel Split Variational Inequality Problems · 2026 · DOIFuture research can focus on testing the approach on other types of constrained optimization problems. Future research can focus on developing a more comprehensive comparison with other solvers. Future research can focus on improving the computational complexity of the approach.
The gap in existing research is the lack of efficient and exact penalty function approaches for solving equality constrained optimization problems over a convex set. The gap is identified as the need for a new approach that can efficiently solve the problem. The gap is also identified as the lack of a comprehensive comparison with other solvers.
The paper suggests future research directions in the development of efficient algorithms for solving vector optimization problems. The study proposes further investigation into the applications of the paper's results in various fields.
On vector variational inequalities and vector optimization problems using strongly convex function · 2026 · DOIThe study recognizes the need to establish relationships among the solutions of Stampacchia and Minty type vector variational inequalities and an efficient minimizer of vector optimization problems.
On vector variational inequalities and vector optimization problems using strongly convex function · 2026 · DOIThe lack of efficient algorithms for solving variational inclusions in Banach spaces. The need for a rigorous basis for establishing the convergence of iterative schemes. The gap in incorporating existing results as special cases.
Graph Convergence Approach of Resolvent Operators to Approximate the Solutions of General Class of Variational Inclusions · 2026 · DOIThe lack of a priori estimates for the solution u. The difficulty of proving the existence of multi-peak solutions.
The paper also identifies challenges due to the complexity of the systems and the need for new control strategies.
Geometric Stability Analysis for Differential Inclusions Governed by Maximally Monotone Operators · 2026 · DOIThe paper suggests future research directions, including the extension of the approach to systems with weaker boundedness or regularity assumptions. The paper also suggests the application of the approach to other types of systems and the development of new control strategies based on the stability analysis.
Geometric Stability Analysis for Differential Inclusions Governed by Maximally Monotone Operators · 2026 · DOIBut, to the best of our knowledge, they still require some extra conditions, including but not limited to the assumption that the abstract set is the whole space.
In a finite-dimensional space setting, we proved the convergence of the whole trajectory to a primal-dual solution under α ≥ 3, without assuming the Lipschitz continuity of ∇f. This extends the trajectory convergence result of Boţ and Nguyen, where the analysis requires Lipschitz continuity of ∇f and the noncritical condition α > 3.
Trajectory convergence and $o(t^{-2})$ rates for Nesterov accelerated primal-dual dynamics without Lipschitz gradient assumption · 2026The results are limited to the context of fully nonlinear second-order equations - The viscosity operator F needs to satisfy certain conditions, such as monotonicity and a Lipschitz condition in the matrix variable - The method does not provide explicit constants in all…
Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton–Jacobi equations via nonlinear convolutions · 2026 · DOIInvestigating the convergence rates for more general classes of equations - Extending the results to other types of approximations - Studying the regularizing effects of the Hamiltonian in more…
Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton–Jacobi equations via nonlinear convolutions · 2026 · DOIInvestigate the necessity of the innate regularity assumption, - Explore the application of the results to other domains, - Study the behavior of the sequence (x n) n∈N in different…
Extending Meshulam’s Result on the Boundedness of Orbits of Relaxed Projections onto Affine Subspaces from Finite to Infinite-Dimensional Hilbert Spaces · 2026 · DOISeveral directions deserve further investigation, - Studying acceleration in coupled primal-dual…
Nesterov acceleration for strongly convex-strongly concave bilinear saddle point problems: Discrete and continuous-time approaches · 2026 · DOIWe further introduced a complementary “anisotropic” notion of firm nonexpansiveness, which allowed us to reinterpret the conjugate duality gap documented in from an operator perspective, as well as in terms of convexity of a function and of its envelope, and to establish a novel characterization of relative smoothness for convex functions under minimal assumptions.
Bregman Level Proximal Subdifferentials and New Characterizations of Bregman Proximal Operators · 2026 · DOI
Most-cited papers in Optimization and Variational Analysis
- Sufficient Conditions in Optimal Control Theory · International Economic Review · 1977 · 214 citations
- The principle of minimum differentiation revisited · European Economic Review · 1984 · 125 citations
- Hessian measures I · Topological Methods in Nonlinear Analysis · 1997 · 110 citations
- The Lagrange Multiplier Rule · American Mathematical Monthly · 1973 · 85 citations
- Hyperbole, extreme case formulation · Journal of Pragmatics · 2004 · 67 citations
- On the consistency of cardinal direction constraints · Artificial Intelligence · 2004 · 61 citations
- First Noether-type theorem for the generalized variational principle of Herglotz · Topological Methods in Nonlinear Analysis · 2002 · 60 citations
- PARALLEL EXTRAGRADIENT-PROXIMAL METHODS FOR SPLIT EQUILIBRIUM PROBLEMS · Mathematical Modelling and Analysis · 2016 · 50 citations
- Lavrentiev Regularization and Balancing Principle for Solving Ill-Posed Problems with Monotone Operators · Computational Methods in Applied Mathematics · 2010 · 44 citations
- Unpacking Burt’s constraint measure · Social Networks · 2020 · 41 citations
Most recent work
- An extrapolated projection and contraction algorithm with past iterates · Numerical Algorithms · 2026
- Golden Ratio Algorithm with Inertia for Non-Lipschitz Variational Inequalities · Journal of Optimization Theory and Applications · 2026
- Efficient Halpern inertial extragradient-based algorithms for solving constrained pseudomonotone VIP systems and their application in blurred image recovery · Communications in Nonlinear Science and Numerical Simulation · 2026
- Remarks on the rate of convergence of the vanishing viscosity process of Hamilton–Jacobi equations · Nonlinearity · 2026
- Pareto Optimality of Critical State Vectors in Vector Sets with State Attributes · Journal of Optimization Theory and Applications · 2026
- An inertial reflected-forward-backward splitting method for monotone inclusions with improved step size · Advances in Computational Mathematics · 2026
- Accelerating diagonal methods for bilevel optimization: Unified convergence via continuous-time dynamics · Mathematical Programming · 2026
- Finite-time and fixed-time stable dynamical systems for solving inverse quasi-variational inequality problems · Optimization · 2026
- On the Complexity of Finding Locally Optimal Solutions in Bilevel Linear Optimization · Operations Research · 2026
- Visualization of fractional reverse inequalities involving interval-valued convexity and their application · Journal of Computational and Applied Mathematics · 2026
Find a gap in your own Optimization and Variational Analysis sub-topic
This page shows what the Optimization and Variational Analysis literature already flags as unresolved. To narrow it to your specific question, search the Research Gap Finder: the search is free with a free account and lists the papers closest to your topic first. Unlocking that topic (50 credits, charged once) fills the comparison table from our 4.5M-paper local library and writes the gaps from its rows.
Open the Research Gap Finder →