A wide range of optimization and control problems can be directly or indirectly formulated as constrained nonlinear equations, and the derivative-free conjugate gradient (CG) projection method is recognized as one of the most effective algorithms for solving constrained nonlinear equations of this type. However, algorithms designed under conventional line search criteria only attain global convergence with an iteration complexity of 𝒪(1/ 4 k) under standard assumptions, and the existing criteria suffer from insufficient diversity. To mitigate these drawbacks, this paper proposes an improved self-adaptive derivative-free CG projection method based on the memoryless BFGS method for constrained nonlinear equations. Specifically, a novel conjugate parameter is derived via the memoryless BFGS method combined with a truncation technique, and a family of line search criteria is developed by introducing a new self-adaptive factor. Theoretical analysis demonstrates that the proposed algorithm ensures convergence without requiring the Lipschitz continuity of the objective mapping or reliance on a specific line search criterion. Under the Lipschitz continuity condition, the algorithm achieves a faster convergence rate with an improved iteration complexity of 𝒪(1/√(k)) . Finally, large-scale numerical experiments on convex constrained nonlinear equations, along with applications in image restoration and sparse signal recovery, validate that the proposed method exhibits superior competitiveness and promising application potential.
In this paper we investigate geometric and analytical properties of p-convexity for sets in ℝ^n , with 0
This paper studies mathematical programs with second-order cone complementarity constraints (SOCMPCCs), which generalize classical mathematical programs with complementarity constraints. For such cone-constrained optimization problems, due to inherent complementarity structure, standard constraint qualifications (such as Robinson’s CQ) fail to hold at any feasible point. By exploiting the spectral decomposition of the second-order cone and the concept of conic linear independence, we first introduce two weak constraint qualifications, termed SOCMPCC-weak-nondegeneracy condition and SOCMPCC-weak-Robinson’s CQ. We show that these conditions are strictly weaker than both SOCMPCC-nondegeneracy condition and SOCMPCC-LICQ. Furthermore, we propose two weak constant rank-type constraint qualifications, namely SOCMPCC-weak-CRCQ and SOCMPCC-weak-CPLD. For these four newly introduced constraint qualifications, we clarify their implication relationships both among themselves and with existing constraint qualifications mentioned above. Finally, we investigate their effectiveness in characterizing stationarity properties of SOCMPCCs and establish that any local minimizer is K-/S-/M-/C-stationary under these conditions.
In this paper, we derive first and second-order optimality conditions of KKT type for locally optimal solutions to a class of multiobjective optimal control problems with endpoint constraints and mixed pointwise constraints. We give some sufficient conditions for normality of multipliers. We show in particular that if the linearized system is controllable or if suitable constraint qualifications hold, then the multiplier corresponding to the objective function is nonzero. We illustrate the theoretical results with an application to sustainable energy management in smart grids.
We present new results on computing the optimal values of a real polynomial via sum of squares approximation. By making a change of variables, we can transform the global optimization problem into one with compact constraints, thereby reducing the degree of the objective function. In most cases, the degree of the objective polynomial after the change of variables is significantly lower than that of the original. A limitation of this approach is that the objective polynomial must be nondegenerate.
In our previous work [8], we proposed two novel mechanisms for generating closed convex cones: one based on systems of inequalities and the other on support functions, together with explicit characterizations of the associated dual and polar cones. Building upon these results, the present paper develops a strengthened and more systematic approach for deriving dual and polar cones from given convex cones. In addition, we establish a significantly extended and unified framework for the construction of convex cones, which broadens the scope of applicability of existing methods and provides greater structural flexibility.
The bi-level form of triple decomposition (TriD) of a third-order tensor has shown promising performance in data representation. In many practical applications, the generated tensor data are non-negative, and the observed data usually lie in or near some low-dimensional subspace. However, TriD does not account for the non-negativity constraint or the local geometric structure of the data. In this paper, we propose a graph regularized non-negative bi-level form of triple decomposition (GNBTD) model based on TriD. The model can extract the low-dimensional part-based representations while maintaining geometrical information from high-dimensional tensor data. In addition, an optimization algorithm based on an accelerated proximal gradient descent method is derived for the GNBTD model, and the convergence of the algorithm is discussed. Experiment results on both synthetic and real data show that our method is efficient and competitive.
Combinatorial optimization problems (COPs) pose a challenge to solve due to their NP-hard nature, motivating the development of metaheuristic methods. However, existing methods often fail to locate the globally optimal solution due to the lack of population diversity. Niching methods, on the other hand, can preserve population diversity, allowing for multiple solutions in a single run, increasing the chance of finding globally optimal solutions, and enabling decision-makers to choose their preferred solution. Although some niching methods have been modified and applied to small-scale COPs, they have inherent difficulties that limit their ability to provide high-quality solutions. To address this limitation, this paper proposes a k-nearest neighbor and Hamming distance-based niching (k-NNHD) scheme for COPs. The k-NNHD method distributes at most k individuals evenly among niches, ensuring that the members of one niche cannot form another and preventing them from leaving their desired niche. To find a promising solution within the niche while balancing exploration and exploitation, the k-NNHD scheme is implemented using a time-dependent inertia weight (TDIW)-based local best binary particle swarm optimization (lbest-BPSO) method. Numerical results show that the k-NNHD method outperforms well-known global optimization methods in solving two sets of 0-1 knapsack problem instances and two collections of multidimensional 0-1 knapsack problem instances characterized by multiple optimal solutions. Moreover, the k-NNHD method also outperforms well-known niching methods in solving two sets of multimodal instances of the 0-1 knapsack problem and one set of traveling salesman problem instances, which is a representative combinatorial optimization problem.
Many algorithms in convex optimization and variational analysis can be analyzed using Fejér monotone sequences. In 2024, Behling, Bello-Cruz, Iusem, Liu, and Santos introduced a new, more general notion: Fejér* monotonicity. They obtained basic results and discussed applications in optimization. In this work, we complement Behling et al.’s work by presenting a thorough study of Fejér* monotonicity. We reveal striking similarities and differences between these notions, including descriptions of the maximal Fejér* set and the importance of the relative interior. Moreover, we touch upon Opial sequences. Throughout this paper, we provide several limiting examples and counterexamples.
We present an interpolation scheme that involves post critically finite iterated function systems. More precisely, while the classical fractal interpolation function, introduced by M. Barnsley, is the attractor of an iterated function system comprising Banach contractions on the metric space [x_0,x_N]× K (where N∈ℕ , N≥ 2 , {(x_i,y_i)∈ℝ^2| i∈{0,1,...,N}} is a data set such that x_0
The paper investigates a singular perturbation for a class of evolution differential variational-hemivariational inequalities. We examine a complex system where a nonlinear parabolic equation interacts with second-order variational-hemivariational inequality involving singular acceleration perturbation. First we establish the solvability in the weak sense of the singularly perturbed system. We then carry out a formal asymptotic expansion as the small parameter vanishes. The analysis confirms that the limit solves a problem in which the second-order perturbation effect is eliminated. Finally, we provide an application to a quasistatic model in contact mechanics for a viscoelastic material with a thermal effect. For the latter, we derive conclusions of weak solvability by the vanishing perturbation approach.
This paper mainly focuses on linear quadratic optimal control problem of multi-dimensional fractional differential equations based on uncertainty theory, which is a significant tool for modeling belief degrees. First, a class of multi-dimensional linear uncertain fractional differential equations is solved analytically by solving auxiliary multi-dimensional linear uncertain differential equations of integer order. Then an expected value model for uncertain fractional LQ optimal control problems is considered and the feedback form of its optimal control solution is obtained by dynamic programming. Moreover, a parametric optimal control model is considered for simplifying the expression of optimal control, and an approximation method is presented to solve the model based on the equation of optimality. As an application, an optimal control problem of unmanned aerial vehicle attitude is given, and numerical experiments are carried out to demonstrate the practicality of the proposed model and the efficiency of the proposed approximation method.
This paper introduces a novel inertial subgradient extragradient algorithm for solving pseudo-monotone variational inequalities in real Hilbert spaces. The proposed method combines three key innovations: (i) a three-point adaptive inertial extrapolation scheme that leverages information from both past iterates and auxiliary points to accelerate convergence; (ii) a fully adaptive step-size rule that operates without prior knowledge of the Lipschitz constant of the operator, enhancing practical applicability; and (iii) a viscosity regularization technique that guarantees strong convergence to a specific solution. Notably, the algorithm requires only a single evaluation of the operator per iteration, significantly reducing computational cost compared to existing inertial extragradient methods. Under standard assumptions, pseudo-monotonicity, and Lipschitz continuity of the operator, we establish strong convergence of the generated sequence to a unique solution. Furthermore, we derive explicit best-iterate convergence rates of order O(log k/k) for the residual terms. Numerical experiments on benchmark problems, including affine monotone operators, non-monotone nonlinear operators in both finite and infinite dimensions, and portfolio optimization, demonstrate that the proposed algorithm consistently outperforms several state-of-the-art methods in terms of iteration count and computational time.
This paper introduces a first-order Clarke-type directional derivative for possibly nonsmooth functions without assuming local Lipschitz continuity, together with a second-order counterpart for Fréchet differentiable functions whose gradients are locally stable. We establish their fundamental properties, basic calculus rules, and generalized forms of the mean value theorem and Taylor expansion. These tools are applied to characterize convexity, strong convexity, and quasiconvexity, thereby extending several classical results beyond the standard Lipschitz or C^1,1 framework. We also derive first-order conditions for local sharp minimizers and second-order conditions for local strong minimizers under these assumptions. Several examples illustrate the framework’s applicability and advantages.
We introduce the Soft Quantum Kernel, a one-parameter family of positive-semidefinite kernels indexed by a softness parameter α∈ [0,1] and obtained by replacing the pure-state density operator of a standard quantum kernel with the output of a depolarizing channel. The construction traces a convex chord on the manifold of density operators between the pure-state kernel and the maximally mixed state, and admits a closed-form expression that separates an informative component, controlled by α , from a uniform regularizing floor. The kernel is shown to satisfy the Mercer property, and an exact bias–variance decomposition of the associated kernel-ridge estimator is derived. The effective degrees of freedom depend continuously on α , so that the joint minimization of the expected prediction error over (α ,λ ) is a biconvex program admitting at least one interior stationary point under standard regularity, in which α acts as a spectral regularizer complementary to the Tikhonov penalty. The softness parameter thus admits a transparent statistical interpretation: it shrinks the kernel spectrum multiplicatively while adding a rank-one component, so that the Soft Quantum Kernel is best read as an interpretable spectral regularization scheme rather than as a new kernel class. A computationally free rule fixes α from the normalized Shannon entropy of an auxiliary uncertainty distribution and provides a principled entry point for cross-validated refinement. A multi-block extension is cast as a kernel-alignment problem over the unit simplex and admits a closed-form solution in the interior case. The framework is connected to Riemannian optimization on the cone of positive-definite matrices via the Bures–Wasserstein metric, linking the present results to geometric algorithms recently developed in this journal. Hardware validation on a superconducting quantum processor confirms that a Frobenius perturbation of the Gram matrix of order one fifth produces a downstream area-under-curve shift below one percent. We stress that the main experimental claim of the paper concerns the stability of the estimator under physical noise rather than its predictive accuracy; a Monte Carlo study on simulated data, including classical kernel and ensemble baselines, noise-injection experiments, and sensitivity analyses with respect to the initialization and the entropy binning, corroborates this claim in a controlled environment.
We study the split common solution problem with multiple output sets. In order to approximate a solution, we first introduce a new cyclic algorithm and establish its weak convergence. Then, in order to obtain strong convergence, we propose two additional algorithms by combining our first algorithm with hybrid or shrinking projection methods. Addressing related problems, we also present several corollaries of our main theorems.
This paper develops two general inertial smoothing Bregman proximal gradient algorithms to tackle the Capped- ℓ _1 group sparse regularization problem with a convex nonsmooth loss function. The first algorithm incorporates two distinct extrapolation terms into the gradient and proximal steps, with the parameters complying with certain constraints, while the other further employs a line search scheme in which the parameters are adaptively updated. We prove that any two accumulation points of the relevant subsequence share the same support set, and the zero groups can be exactly identified within a finite number of iterations. Moreover, any accumulation point is a lifted stationary point of the regularization model. Numerical experiments on several test problems demonstrate the efficiency of the proposed algorithms.
This paper investigates a class of double phase inclusion problems with variable exponents and mixed boundary conditions on bounded Lipschitz domains. The right-hand side of the problem involves both Clarke subdifferentials, describing nonconvex constraints, and convex subdifferentials, corresponding to convex constraints. By constructing a total functional associated with the problem that combines convex and locally Lipschitz components, and by applying Ekeland’s variational principle together with Yosida approximation techniques and subdifferential calculus, we establish the existence of nontrivial bounded weak solutions under suitable assumptions. Our results extend the ones established in [J. Differential Equations 316 (2022), 249-269].
