A parametric linearizing approach for quadratically inequality constrained quadratic programs
-
Hongwei Jiao
and
Rongjiang Chen
Abstract
In this paper we propose a new parametric linearizing approach for globally solving quadratically inequality constrained quadratic programs. By utilizing this approach, we can derive the parametric linear programs relaxation problem of the investigated problem. To accelerate the computational speed of the proposed algorithm, an interval deleting rule is used to reduce the investigated box. The proposed algorithm is convergent to the global optima of the initial problem by subsequently partitioning the initial box and solving a sequence of parametric linear programs relaxation problems. Finally, compared with some existing algorithms, numerical results show higher computational efficiency of the proposed algorithm.
1 Introduction
In this paper we consider the following quadratically inequality constrained quadratic programs:
where
In last decades, for the problem (QICQP) and its special cases many methods have been developed and described in the existent literature. For example, semi-definite relaxation method [1], reformulation-convexification approach [2], branch-and-reduce approaches [3, 4, 5, 6, 7], approximation algorithms [8, 9, 10], simplicial branch-and-bound method [11], branch-and-cut method [12], rectangle branch-and-bound algorithms [13,14], robust solution approach [15], and so on. In addition, some algorithms for geometric programming [16, 17, 18, 19, 20] and multiplicative (or fractional) programming [21, 22, 23, 24, 25] also can be used to solve the (QICQP). Although these methods can be used to solve the investigated problem (QICQP) or its special forms, less work has been still done for globally solving the investigated quadratically inequality constrained quadratic programs.
In this paper, we will present a new branch-and-bound algorithm for globally solving the (QICQP). Firstly, we present a new parametric linearizing technique. By utilizing this method, we can convert the (QICQP) into a parametric linear programs relaxation problem, which can be used to compute the lower bounds of the optimal values of the initial problem (QICQP) and its subproblems. Secondly, based on the branch-and-bound framework, by successive partitioning of the initial box and by solving those derived parametric linear programs relaxation problems, a new branch-and-bound algorithm is designed for globally solving the (QICQP). Thirdly, to accelerate the computational efficiency of the proposed branch-and-bound algorithm, an interval deleting rule is used to reduce the investigated box. Fourthly, the proposed algorithm is convergent to the global optima of the initial problem (QICQP) by successively partitioning of the initial box and by solving those derived parametric linear programs relaxation problems. Finally, compared with some existent algorithms, numerical results demonstrate the computational efficiency of the proposed algorithm.
The remaining sections of this article are organized as follows. First of all, we present a new parametric linearizing technique for deriving the parametric linear programs relaxation problem of the (QICQP) in Section 2. Secondly, based on the branch-and-bound framework, by combing the derived parametric linear programs relaxation problem with the interval deleting rule, a branch-and-bound algorithm is established for globally solving the (QICQP) in Section 3. Thirdly, compared with the existent methods, some numerical examples in existent literatures are used to verify the computational efficiency of the proposed algorithm in Section 4. Finally, some concluding remarks are presented.
2 New parametric linearizing approach
In this section, we propose a new parametric linearizing approach for deriving the parametric linear programs relaxation problem of the (QICQP). The detailed parametric linearizing approach is presented as follows:
Assume that Z = {(z1, z2, …, zn)T ∈ Rn : lj ≤ zj ≤ uj, j = 1, …, n} ⊆ Z0, λ = (λjk)n × n ∈ Rn × n is a symmetric matrix, and λjk ∈ {0, 1}. For convenience, for any z ∈ Z, for any k ∈ {1, 2, …, n}, some expressions are introduced as follows:
Obviously, we have zk(0) = lk, zk(1) = uk.
Theorem 2.1
For any k ∈ {1, 2, …, n}, for any z ∈ Z, consider the functions hkk(z), hkk(z, Z, λkk) and hkk(z, Z, λkk), then, the following conclusions hold:
and
Proof
By the mean value theorem, for any z ∈ Z, there exists a point ξk = α zk+(1 − α)zk(λkk), where α ∈ [0, 1], such that
If λkk = 0, then we have
If λkk = 1, then it follows that
Thus, we can get that
Similarly, if λkk = 0, then we have
If λkk = 1, then it follows that
Thus, we can get that
Therefore, for any z ∈ Z, we have that
Since
we have
Also since
Therefore, it follows that
The proof is completed. □
Without loss of generality, for any z ∈ Z, for any j ∈ {1, 2, …, n}, k ∈ {1, 2, …, n}, j ≠ k, we define
Obviously, we have (zj − zk)(0) = lj − uk, (zj − zk)(1) = uj − lk.
In a similar way as in Theorem 2.1, we can get the following Theorem 2.2:
Theorem 2.2
For each j = 1, 2, …, n, k = 1, 2, …, n, for any z ∈ Z, we have:
The following inequalities hold:
The following limitations hold:
Proof
From the inequality (1), replacing λkk by λjk, and replacing zk by zj, we can get that
From the inequality (1), replacing λkk by λjk, we can get that
From (1), replacing λkk and zk by λjk and (zj − zk), respectively, we can get that
From the limitations (2) and (3), replacing λkk by λjk, it follows that
and
By the limitations (2) and (3), replacing λkk and zk by λjk and (zj − zk), respectively, we can get that
and
The proof is completed. □
Without loss of generality, for any z ∈ Z, for any j ∈ {1, 2, …, n}, k ∈ {1, 2, …, n}, j ≠ k, define
Theorem 2.3
For each k = 1, 2, …, n, consider the functions hjk(z, Z, λjk), hjk(z) and hjk(z, Z, λjk), then, for any z ∈ Z, we have the following conclusions:
and
Proof
First of all, from the conclusions (i) of Theorem 2.2, it follows that
and
Secondly, from the inequalities (4) and (5), we have
Thus, we can get that lim∥u − l∥ → 0[hjk(z) − hjk(z, Z, λjk)] = 0.
Also from the inequalities (4) and (5), we get that
Thus, it follows that lim∥u − l∥ → 0[hjk(z, Z, λjk) − hjk(z)] = 0. □
Without loss of generality, for any Z = [l, u] ⊆ Z0, for any parameter matrix λ = (λjk)n × n, for any z ∈ Z and i ∈ {0, 1, …, m}, we let
Theorem 2.4
For ∀ z ∈ Z = [l, u] ⊆ Z0, for any given parameter matrix λ = (λjk)n × n, for each i = 0, 1, …, m, we have the following conclusions:
and
Proof
First of all, from the inequalities (1) and (6), for any j, k ∈ {1, …, n}, we have
By the above inequalities (9) and (10), for any z ∈ Z ⊆ Z0, we can get that
Therefore, we have
Secondly,
From the limitations (2), (3), (7) and (8), lim∥u − l∥ → 0[hkk(z) − hkk(z, Z, λkk)] = 0, lim∥u − l∥ → 0[hkk(z, Z, λkk) − hkk(z)] = 0, lim∥u − l∥ → 0[hjk(z) − hjk(z, Z, λjk)] = 0 and lim∥u − l∥ → 0[hjk(z, Z, λjk) − hjk(z)] = 0.
Therefore, it follows that
Similarly, we can prove that
The proof is completed. □
By Theorem 2.4, we can construct the parametric linear programs relaxation problem (PLPRP) of the (QICQP) over Z as follows:
where
Based on the former parametric linearizing technique, each feasible solution of the (QICQP) must be also feasible to the (PLPRP) in the sub-region Z; and the minimum value of the (PLPRP) must be less than or equal to that of the (QICQP) in the sub-region Z. Hence, the (PLPRP) offers a reliable lower bound for the minimum value of the (QICQP) in the sub-region Z. In addition, Theorem 2.4 ensures that the optimal solution of the (PLPRP) will sufficiently approximate the optimal solution of the (QICQP) as ∥u − l∥ → 0, and this guarantees the global convergence of the proposed algorithm.
3 Branch-and-bound global optimization algorithm
In this section, a new branch-and-bound global optimization algorithm is proposed for solving the (QICQP). In this algorithm, there are the following several important techniques: branching, bounding the lower bound, bounding upper bound and interval deleting.
Branching: The branching step will generate a more refined box partition. Here we choose a typical box-bisection method, which is sufficient to ensure the global convergence of the proposed branch-and-bound method. For any selected box Z′ = [l′, u′] ⊆ Z0. Set η ∈ arg max
Bounding the lower bound: For each sub-box Z ⊆ Z0, which has not been deleted, the bounding the lower bound step needs to solve the parametric linear programs relaxation problem over each sub-box, and denote by LBs = min{LB(Z)|Z ∈ Ωs}, where Ωs denotes the set of sub-box which has not been deleted after s iteration.
Bounding the upper bound: The bounding upper bound step needs to judge the feasibility of the midpoint of each investigated sub-box Z and the optimal solution of the (PLPRP) over the investigated sub-box Z, where Z ∈ Ωs. In addition, we need to calculate the objective function values of each known feasible solutions for the (QICQP), and denote by UBs = min{H0(z) : z ∈ Θ} the best upper bound, where Θ is the known feasible point set.
Interval deleting: To improve the convergent speed of the branch-and-bound algorithm, an interval deleting rule is introduced as follows. For convenience, for any z ∈ Z, i ∈ {0,1, …, m}, q ∈ {1, …, n}, and denote by HUB the current upper bound of the (QICQP), we let
Theorem 3.1
For any investigated sub-boxZ = (Zj)1×n ⊆ Z0, we have the following conclusions:
IfRLB0(λ) > HUB, then the whole investigated sub-boxZshould be deleted.
IfRLB0(λ) ≤ HUB, then: for anyq ∈ {1,2, …, n}, ifα0q(λ) > 0, the intervalZqshould be replaced by
IfRLBi(λ) > bifor somei ∈ {1, …, m}, then the whole investigated sub-boxZshould be deleted.
IfRLBi(λ) ≤ bifor eachi ∈ {1, …, m}, then, for anyq ∈ {1,2, …, n}, ifαiq(λ) > 0, the intervalZqcan be replaced by
Proof
In a similar way as in the proof of Theorem 3 in [14], we may draw the conclusions for Theorem 3.1, so here it is omitted.
From Theorem 3.1, we can construct an interval deleting step to compress the investigated box for improving the convergent speed of the proposed branch-and-bound algorithm.
3.1 New branch-and-bound algorithm
For any sub-box Zs ⊆ Z0, we denote by LB(Zs) the optimal value of the (PLPRP) over the sub-box Zs, and denote by zs = z(Zs) the optimal solution of the (PLPRP) over the sub-box Zs. Based on the branch-and-bound framework, combining the former branching step, bounding the lower bound step, bounding upper bound step and interval deleting step together, a new branch-and-bound algorithm is designed as follows.
Branch-and-Bound Algorithm Steps
Initializing step. Given the initial convergent error ϵ, the initial randomly generated parameter matrix λ.
Solve the (PLPRP) over the initial box Z0 to obtain its optimal solution z0 and optimal value LB(Z0), denote by the initial lower bound LB0 = LB(Z0). If z0 is a feasible solution of the (QICQP), we denote by the initial upper bound UB0 = H0(z0). Otherwise, we denote by the initial upper bound UB0 = + ∞.
If UB0 − LB0 ≤ ϵ, the proposed algorithm terminates, z0 is a global ϵ-optimal solution of the initial problem (QICQP). Otherwise, set Ω0 = {Z0}, Λ = ∅, s = 1.
Branching step. Let UBs = UBs−1. Partition the investigated sub-box Zs−1 into two sub-boxes Zs,1, Zs,2 by the selected branching rule, and denote by Λ = Λ ∪ {Zs−1} the set of the deleting sub-boxes.
Interval deleting step. For each investigated sub-box Zs,t, t = 1, 2, use the former interval deleting rule to compress the investigated sub-box, still denote by Zs,t the remaining sub-box.
Bounding the lower bound step. For each remaining sub-box Zs,t, where t = 1, 2, solve the (PLPRP) over Zs,t to obtain its optimal solution zs,t and optimal value LB(Zs,t), and let Ωs = {Z|Z ∈ Ωs−1 ∪ {Zs,1, Zs,2}, Z ∉ Λ} and LBs = min{LB(Z)|Z ∈ Ωs}.
Bounding the upper bound step. For each sub-box Zs,t, if its midpoint zmid is the feasible point of the initial problem (QICQP), let Θ := Θ ∪ {zmid}, denote by the new upper bound UBs = minz∈ΘH0(z); if the optimal solution zs,t of the (PLPRP) is the feasible point of the initial problem (QICQP), denote by the new upper bound UBs = min{UBs, H0(zs,t)}, and denote by zs the best existent feasible point such that UBs = H0(zs).
Terminating judgement step. If UBs − LBs ≤ ϵ, the proposed algorithm terminates, zs is a global ϵ-optimal solution of the initial problem (QICQP). Otherwise, denote by s = s + 1, and go to the Branching step.
3.2 Global convergence of the proposed algorithm
Without loss of generality, we assume that v is the global optimal value of the initial problem (QICQP). If the proposed algorithm terminates after s finite iterations, where s is a finite number such that s ≥ 0, then it follows that
From the bounding the upper bound step of the proposed algorithm, we know that there must exist a feasible point zs of the initial problem (QICQP) such that
By the branch-and-bound structure of the proposed algorithm, we have
Combining the above several inequalities together, it follows that
Therefore, zs is an ϵ-global optimal solution of the initial problem (QICQP).
If the proposed algorithm does not terminate after finite iterations, for this case, the detailed convergent conclusions are given as follows.
Theorem 3.2
If the proposed algorithm does not terminate after finite iterations, then it will generate an infinite partitioning sequence {Zs} of the initial boxZ0, and any accumulation point of the sequence {Zs} will be a global optimum solution of the initial problem (QICQP).
Proof
First of all, in the proposed algorithm the selected branching method is the bisection of box, so that the branching process is exhaustive, that is to say, the branching step will ensure that the intervals of all variables tend to 0, i.e., ‖u – l‖ → 0.
Secondly, from Theorem 2.4, the optimal solution of the (PLPRP) will sufficiently approximate the optimal solution of the (QICQP) as ‖u – l‖ → 0, and this ensures that the limitation lims→∞(UBs − LBs) = 0 holds. So that the bounding operation is consistent.
Thirdly, in the proposed algorithm the subdivided box which attains the actual lower bound is selected for further partition at the later immediate iteration, so that the used selecting operation is bound improving.
From [26, Theorem IV.3], the sufficient condition of global convergence of the branch-and-bound algorithm is that the branching method is exhaustive, the bounding method is consistent and the selecting method is improvement, therefore, the proposed algorithm is convergent to the global optimal solution of the initial (QICQP). □
4 Numerical experiments
Given the convergent error ϵ = 10−6 and the parameter matrix λ = (λjk)n×n ∈ Rn×n, where λjk ∈ {0, 1}, compared with the existing methods, several numerical examples in existing literature are tested on microcomputer, the procedure is coded in C++ software, the parametric linear programs relaxation problems are solved by the simplex method. These examples and their numerical results are listed as follows. In the following Tables 1 and 2, the number of iteration and running time in seconds for the algorithm are represented by “Iteration” and “Time(s)”, respectively.
Numerical comparisons for Examples 4.1-4.7
| Example | Refs. | Optimal value | Optimal solution | Iteration | Time(s) |
|---|---|---|---|---|---|
| 1 | our | 1.177124344 | (1.177124344, 2.177124344) | 22 | 0.0103 |
| [16] | 1.177124327 | (1.177124327, 2.177124353) | 434 | 1.0000 | |
| 2 | our | -0.999999106 | (2.000000, 1.000000) | 21 | 0.0079 |
| [16] | -1.0 | (2.000000, 1.000000) | 24 | 0.0129 | |
| 3 | our | 6.778046494 | (2.000000000, 1.666747279) | 12 | 0.0039 |
| [4] | 6.777778340 | (2.000000000, 1.666666667) | 30 | 0.0068 | |
| [5] | 6.777782016 | (2.000000000, 1.666666667) | 40 | 0.0320 | |
| [17] | 6.7780 | (2.00003, 1.66665) | 44 | 0.1800 | |
| 4 | our | 0.500000000 | (0.500000000, 0.500000000) | 25 | 0.0070 |
| [5] | 0.500004627 | (0.5, 0.5) | 34 | 0.0560 | |
| [14] | 0.500000442 | (0.500000000, 0.500000000) | 37 | 0.0193 | |
| [17] | 0.5 | (0.5, 0.5) | 91 | 0.8500 | |
| [18] | 0.5 | (0.5, 0.5) | 96 | 1.0000 | |
| 5 | our | 118.392375925 | (2.560178568, 3.125000000) | 46 | 0.0294 |
| [4] | 118.383672050 | (2.555409888, 3.130613160) | 49 | 0.0744 | |
| [6] | 118.383756475 | (2.5557793695, 3.1301646393) | 210 | 0.7800 | |
| [14] | 118.383671904 | (2.555745855, 3.130201688) | 59 | 0.0385 | |
| 6 | our | -1.162882315 | (1.499977112, 1.5) | 37 | 0.0756 |
| [19] | -1.16288 | (1.5, 1.5) | 84 | 0.1257 | |
| 7 | our | -11.363636364 | (1.0, 0.181815071, 0.983332741) | 98 | 0.1672 |
| [14] | -11.363636364 | (1.0, 0.181818470, 0.983332113) | 420 | 0.2845 | |
| [20] | -10.35 | (0.998712,0.196213,0.979216) | 1648 | 0.3438 |
Numerical results for Example 4.8
| Refs. | Dimension n | Optimal value | Iteration | Time(s) |
|---|---|---|---|---|
| This paper | 5 | 25.0 | 11 | 0.01632 |
| 10 | 100.0 | 30 | 0.22646 | |
| 20 | 400.0 | 86 | 4.35751 | |
| 30 | 900.0 | 204 | 31.2676 | |
| 40 | 1600.0 | 300 | 80.161 | |
| [3] | 5 | 25.0 | 141 | 10.11 |
| 10 | 100.0 | 283 | 21.86 | |
| 20 | 400.0 | 651 | 47.00 | |
| 30 | 900.0 | 965 | 106.33 | |
| [14] | 5 | 25.0 | 12 | 0.01818 |
| 10 | 100.0 | 32 | 0.30216 | |
| 20 | 400.0 | 88 | 6.01095 | |
| 30 | 900.0 | 206 | 44.4965 | |
| 40 | 1600.0 | 302 | 98.122 |
Example 4.1
([16]).
Example 4.2
([16]).
Example 4.7
Compared with the existing algorithms, the numerical results for examples 1-8 show that the proposed algorithm can be used to globally solve the quadratically inequality constrained quadratic programs with higher computational efficiency.
5 Concluding remarks
In this paper, we propose a new branch-and-bound algorithm for globally solving the quadratically inequality constrained quadratic programs. In this algorithm, we present a new parametric linearizing technique, which can be used to derive the parametric linear programs relaxation problem of the investigated problem (QICQP). To accelerate the computational speed of the proposed branch-and-bound algorithm, an interval deleting rule is used to reduce the investigated box. By subsequently partitioning the initial box and solving a sequence of parametric linear programs relaxation problems, the proposed algorithm is convergent to the global optima of the initial problem (QICQP). Finally, compared with some existing algorithms, numerical results show higher computational efficiency of the proposed algorithm.
Acknowledgement
This paper is supported by the National Natural Science Foundation of China (11671122, 11471102), the China Postdoctoral Science Foundation (2017M622340), the Basic and Frontier Technology Research Program of Henan Province (152300410097), the Cultivation Plan of Young Key Teachers in Colleges and Universities of Henan Province (2016GGJS-107), the Higher School Key Scientific Research Projects of Henan Province (18A110019,17A110021), the Major Scientific Research Projects of Henan Institute of Science and Technology (2015ZD07), the High-level Scientific Research Personnel Project for Henan Institute of Science and Technology (2015037), Henan Institute of Science and Technology Postdoctoral Science Foundation.
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- Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
- Topological entropy for positively weak measure expansive shadowable maps
- Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
- On 𝓠-regular semigroups
- One kind power mean of the hybrid Gauss sums
- A reduced space branch and bound algorithm for a class of sum of ratios problems
- Some recurrence formulas for the Hermite polynomials and their squares
- A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
- On f - prime radical in ordered semigroups
- Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
- Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
- A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
- Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
- The points and diameters of quantales
- Directed colimits of some flatness properties and purity of epimorphisms in S-posets
- Super (a, d)-H-antimagic labeling of subdivided graphs
- On the power sum problem of Lucas polynomials and its divisible property
- Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
- On generalized P-reducible Finsler manifolds
- On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
- On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
- On the different kinds of separability of the space of Borel functions
- Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
- Functional analysis method for the M/G/1 queueing model with single working vacation
- Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
- The existence of solutions to certain type of nonlinear difference-differential equations
- Domination in 4-regular Knödel graphs
- Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
- Algebras of right ample semigroups
- Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
- Nontrivial periodic solutions to delay difference equations via Morse theory
- A note on the three-way generalization of the Jordan canonical form
- On some varieties of ai-semirings satisfying xp+1 ≈ x
- Abstract-valued Orlicz spaces of range-varying type
- On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
- Arithmetic of generalized Dedekind sums and their modularity
- Multipreconditioned GMRES for simulating stochastic automata networks
- Regularization and error estimates for an inverse heat problem under the conformable derivative
- Transitivity of the εm-relation on (m-idempotent) hyperrings
- Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
- Simultaneous prediction in the generalized linear model
- Two asymptotic expansions for gamma function developed by Windschitl’s formula
- State maps on semihoops
- 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
- Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
- New topology in residuated lattices
- Optimality and duality in set-valued optimization utilizing limit sets
- An improved Schwarz Lemma at the boundary
- Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
- Toeplitz matrices whose elements are coefficients of Bazilevič functions
- Epi-mild normality
- Nonlinear elastic beam problems with the parameter near resonance
- Orlicz difference bodies
- The Picard group of Brauer-Severi varieties
- Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
- Weak group inverse
- Infinite growth of solutions of second order complex differential equation
- Semi-Hurewicz-Type properties in ditopological texture spaces
- Chaos and bifurcation in the controlled chaotic system
- Translatability and translatable semigroups
- Sharp bounds for partition dimension of generalized Möbius ladders
- Uniqueness theorems for L-functions in the extended Selberg class
- An effective algorithm for globally solving quadratic programs using parametric linearization technique
- Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
- On categorical aspects of S -quantales
- On the algebraicity of coefficients of half-integral weight mock modular forms
- Dunkl analogue of Szász-mirakjan operators of blending type
- Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
- Global stability of a distributed delayed viral model with general incidence rate
- Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
- Boundary value problems of a discrete generalized beam equation via variational methods
- Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
- Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
- Spectrum of free-form Sudoku graphs
- Regularity of fuzzy convergence spaces
- The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
- On further refinements for Young inequalities
- Pretty good state transfer on 1-sum of star graphs
- On a conjecture about generalized Q-recurrence
- Univariate approximating schemes and their non-tensor product generalization
- Multi-term fractional differential equations with nonlocal boundary conditions
- Homoclinic and heteroclinic solutions to a hepatitis C evolution model
- Regularity of one-sided multilinear fractional maximal functions
- Galois connections between sets of paths and closure operators in simple graphs
- KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
- θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
- An integral that counts the zeros of a function
- On rough sets induced by fuzzy relations approach in semigroups
- Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
- The fourth order strongly noncanonical operators
- Topical Issue on Cyber-security Mathematics
- Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
- Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
- On dynamic network security: A random decentering algorithm on graphs
Articles in the same Issue
- Regular Articles
- Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras
- On a viscous two-fluid channel flow including evaporation
- Generation of pseudo-random numbers with the use of inverse chaotic transformation
- Singular Cauchy problem for the general Euler-Poisson-Darboux equation
- Ternary and n-ary f-distributive structures
- On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics
- Evaluation of integrals with hypergeometric and logarithmic functions
- Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
- Oscillation of first order linear differential equations with several non-monotone delays
- Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
- The log-concavity of the q-derangement numbers of type B
- Generalized state maps and states on pseudo equality algebras
- Monotone subsequence via ultrapower
- Note on group irregularity strength of disconnected graphs
- On the security of the Courtois-Finiasz-Sendrier signature
- A further study on ordered regular equivalence relations in ordered semihypergroups
- On the structure vector field of a real hypersurface in complex quadric
- Rank relations between a {0, 1}-matrix and its complement
- Lie n superderivations and generalized Lie n superderivations of superalgebras
- Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
- Stability problems and numerical integration on the Lie group SO(3) × R3 × R3
- On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
- On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
- A greedy algorithm for interval greedoids
- On nonlinear evolution equation of second order in Banach spaces
- A primal-dual approach of weak vector equilibrium problems
- On new strong versions of Browder type theorems
- A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
- Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
- Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
- Introduction to disoriented knot theory
- Restricted triangulation on circulant graphs
- Boundedness control sets for linear systems on Lie groups
- Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
- Disjointed sum of products by a novel technique of orthogonalizing ORing
- A parametric linearizing approach for quadratically inequality constrained quadratic programs
- Generalizations of Steffensen’s inequality via the extension of Montgomery identity
- Vector fields satisfying the barycenter property
- On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
- Biderivations of the higher rank Witt algebra without anti-symmetric condition
- Some remarks on spectra of nuclear operators
- Recursive interpolating sequences
- Involutory biquandles and singular knots and links
- Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
- Topological entropy for positively weak measure expansive shadowable maps
- Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
- On 𝓠-regular semigroups
- One kind power mean of the hybrid Gauss sums
- A reduced space branch and bound algorithm for a class of sum of ratios problems
- Some recurrence formulas for the Hermite polynomials and their squares
- A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
- On f - prime radical in ordered semigroups
- Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
- Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
- A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
- Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
- The points and diameters of quantales
- Directed colimits of some flatness properties and purity of epimorphisms in S-posets
- Super (a, d)-H-antimagic labeling of subdivided graphs
- On the power sum problem of Lucas polynomials and its divisible property
- Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
- On generalized P-reducible Finsler manifolds
- On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
- On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
- On the different kinds of separability of the space of Borel functions
- Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
- Functional analysis method for the M/G/1 queueing model with single working vacation
- Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
- The existence of solutions to certain type of nonlinear difference-differential equations
- Domination in 4-regular Knödel graphs
- Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
- Algebras of right ample semigroups
- Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
- Nontrivial periodic solutions to delay difference equations via Morse theory
- A note on the three-way generalization of the Jordan canonical form
- On some varieties of ai-semirings satisfying xp+1 ≈ x
- Abstract-valued Orlicz spaces of range-varying type
- On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
- Arithmetic of generalized Dedekind sums and their modularity
- Multipreconditioned GMRES for simulating stochastic automata networks
- Regularization and error estimates for an inverse heat problem under the conformable derivative
- Transitivity of the εm-relation on (m-idempotent) hyperrings
- Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
- Simultaneous prediction in the generalized linear model
- Two asymptotic expansions for gamma function developed by Windschitl’s formula
- State maps on semihoops
- 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
- Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
- New topology in residuated lattices
- Optimality and duality in set-valued optimization utilizing limit sets
- An improved Schwarz Lemma at the boundary
- Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
- Toeplitz matrices whose elements are coefficients of Bazilevič functions
- Epi-mild normality
- Nonlinear elastic beam problems with the parameter near resonance
- Orlicz difference bodies
- The Picard group of Brauer-Severi varieties
- Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
- Weak group inverse
- Infinite growth of solutions of second order complex differential equation
- Semi-Hurewicz-Type properties in ditopological texture spaces
- Chaos and bifurcation in the controlled chaotic system
- Translatability and translatable semigroups
- Sharp bounds for partition dimension of generalized Möbius ladders
- Uniqueness theorems for L-functions in the extended Selberg class
- An effective algorithm for globally solving quadratic programs using parametric linearization technique
- Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
- On categorical aspects of S -quantales
- On the algebraicity of coefficients of half-integral weight mock modular forms
- Dunkl analogue of Szász-mirakjan operators of blending type
- Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
- Global stability of a distributed delayed viral model with general incidence rate
- Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
- Boundary value problems of a discrete generalized beam equation via variational methods
- Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
- Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
- Spectrum of free-form Sudoku graphs
- Regularity of fuzzy convergence spaces
- The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
- On further refinements for Young inequalities
- Pretty good state transfer on 1-sum of star graphs
- On a conjecture about generalized Q-recurrence
- Univariate approximating schemes and their non-tensor product generalization
- Multi-term fractional differential equations with nonlocal boundary conditions
- Homoclinic and heteroclinic solutions to a hepatitis C evolution model
- Regularity of one-sided multilinear fractional maximal functions
- Galois connections between sets of paths and closure operators in simple graphs
- KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
- θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
- An integral that counts the zeros of a function
- On rough sets induced by fuzzy relations approach in semigroups
- Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
- The fourth order strongly noncanonical operators
- Topical Issue on Cyber-security Mathematics
- Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
- Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
- On dynamic network security: A random decentering algorithm on graphs
