A new heuristic approach for the P-median problem

The Euclidean p-median problem is concerned with the decision of the locations for public service centres. Existing methods for the planar Euclidean p-median problems are capable of efficiently solving problems of relatively small scale. This paper proposes two new heuristic algorithms aiming at problems of large scale. Firstly, to reflect the different degrees of proximity to optimality, a new kind of local optimum called level-m optimum is defined. For a level-m optimum of a p-median problem, where m<p, each of its subsets containing m of the p partitions is a global optimum of the corresponding m-median subproblem. Starting from a conventional local optimum, the first new algorithm efficiently improves it to a level-2 optimum by applying an existing exact algorithm for solving the 2-median problem. The second new algorithm further improves it to a level-3 optimum by applying a new exact algorithm for solving the 3-median problem. Comparison based on experimental results confirms that the proposed algorithms are superior to the existing heuristics, especially in terms of solution quality.     

A Lagrangian search method for the <Emphasis Type="Italic">P</Emphasis>-median problem

In this paper, we propose a novel algorithm for solving the classical P-median problem. The essential aim is to identify the optimal extended Lagrangian multipliers corresponding to the optimal solution of the underlying problem. For this, we first explore the structure of the data matrix in P-median problem to recast it as another equivalent global optimization problem over the space of the extended Lagrangian multipliers. Then we present a stochastic search algorithm to find the extended Lagrangian multipliers corresponding to the optimal solution of the original P-median problem. Numerical experiments illustrate that the proposed algorithm can effectively find a global optimal or very good suboptimal solution to the underlying P-median problem, especially for the computationally challenging subclass of P-median problems with a large gap between the optimal solution of the original problem and that of its Lagrangian relaxation.     

The connected <Emphasis Type="Italic">p</Emphasis>-median problem on block graphs

In this paper, we study a variant of the p-median problem on block graphs G in which the p-median is asked to be connected, and this problem is called the connected p-median problem. We first show that the connected p-median problem is NP-hard on block graphs with multiple edge weights. Then, we propose an O(n)-time algorithm for solving the problem on unit-edge-weighted block graphs, where n is the number of vertices in G.     

<Emphasis Type="Italic">p</Emphasis>-Medians and Multi-Medians

The classical p-median problem is discussed, together with methods for its solution. The multi-median problem, a generalization of the p-median problem in which more than one type of facility is allowed, is introduced and methods of solution developed. Numerical results are presented.     

Approximate solution of the <Emphasis Type="Italic">p</Emphasis>-median minimization problem

A version of the facility location problem (the well-known p-median minimization problem) and its generalization—the problem of minimizing a supermodular set function—is studied. These problems are NP-hard, and they are approximately solved by a gradient algorithm that is a discrete analog of the steepest descent algorithm. A priori bounds on the worst-case behavior of the gradient algorithm for the problems under consideration are obtained. As a consequence, a bound on the performance guarantee of the gradient algorithm for the p-median minimization problem in terms of the production and transportation cost matrix is obtained.     

The pos/neg-weighted 2-medians in balanced trees with subtree-shaped customers

This paper deals with the pos/neg-weighted p-median problem on tree graphs where all customers are modeled as subtrees. We present a polynomial algorithm for the 2-median problem on an arbitrary tree. Then we improve the time complexity to O(n log n) for the problem on a balanced tree, where n is the number of the vertices in the tree.     

Heuristic Solution Methods for Two Location Problems with Unreliable Facilities

In this paper, the p-median and p-centre problems are generalized by considering the possibility that one or more of the facilities may become inactive. The unreliable p-median problem is defined by introducing the probability that a facility becomes inactive. The (p, q)-centre problem is defined when p facilities need to be located but up to q of them may become unavailable at the same time. An heuristic procedure is presented for each problem. A rigorous procedure is discussed for the (p, q)-centre problem. Computational results are presented.     

Constructive heuristics for the uncapacitated continuous location-allocation problem

The multisource location-allocation problem in continuous space is investigated. Two constructive heuristic techniques are proposed to solve this problem. Both methods are based on designing suitable schemes for the generation of the initial solutions. The first considers the furthest distance rule and is enhanced by schemes borrowed from tabu search such as constructing the forbidden regions and freeing strategy. The second considers the discrete solutions found when solving the p-median problem. Some results on existing test problems are presented.     

Convergence of greedy algorithm in Walsh system in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

Convergence of the greedy algorithm in Walsh system in L _p , p > 1 is studied. It is proved that there exists a function in L _p , 1 < p < 2, with greedy algorithm not converging in measure to that function. A continuous function with divergent in L _p , p > 2, greedy algorithm is constructed and sufficient conditions for convergence of the greedy algorithm in L _p , p > 1 are given.     

Efficiency of weak greedy algorithms for m-term approximations

We investigate the efficiency of weak greedy algorithms for m-term expansional approximation with respect to quasi-greedy bases in general Banach spaces.We estimate the corresponding Lebesgue constants for the weak thresholding greedy algorithm(WTGA) and weak Chebyshev thresholding greedy algorithm.Then we discuss the greedy approximation on some function classes.For some sparse classes induced by uniformly bounded quasi-greedy bases of L_p,1p∞,we show that the WTGA realizes the order of the best m-term approximation.Finally,we compare the efficiency of the weak Chebyshev greedy algorithm(WCGA) with the thresholding greedy algorithm(TGA) when applying them to quasi-greedy bases in L_p,1≤p∞,by establishing the corresponding Lebesgue-type inequalities.It seems that when p2 the WCGA is better than the TGA.     

Revisiting <Emphasis Type="Italic">k</Emphasis>-sum optimization

In this paper, we address continuous, integer and combinatorial k-sum optimization problems. We analyze different formulations of this problem that allow to solve it through the minimization of a relatively small number of minisum optimization problems. This approach provides a general tool for solving a variety of k-sum optimization problems and at the same time, improves the complexity bounds of many ad-hoc algorithms previously reported in the literature for particular versions of this problem. Moreover, the results developed for k-sum optimization have been extended to the more general case of the convex ordered median problem, improving upon existing solution approaches.     

Simultaneous approximation problems of <Emphasis Type="Italic">p</Emphasis>-adic numbers and <Emphasis Type="Italic">p</Emphasis>-adic knapsack cryptosystems - Alice in <Emphasis Type="Italic">p</Emphasis>-adic numberland

In this paper we construct the multi-dimensional p-adic approximation lattices by using simultaneous approximation problems (SAP) of p-adic numbers and we estimate the l _∞ norm of the p-adic SAP solutions theoretically by applying Dirichlet’s principle and numerically by using the LLL algorithm. By using the SAP solutions as private keys, the security of which depends on NP-hardness of SAP or the shortest vector problems (SVP) of p-adic lattices, we propose a p-adic knapsack cryptosystem with commitment schemes, in which the sender Alice prepares ciphertexts and the verification keys in her p-adic numberland.     

On <Emphasis Type="Italic">c</Emphasis><Superscript>#</Superscript>-normal subgroups infinite groups

A subgroup H of a finite group G is called a c^#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H ∩ K is a CAP-subgroup of G: In this paper, we investigate the influence of fewer c^#-normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.     

On variable step Hermite–Birkhoff solvers combining multistep and 4-stage DIRK methods for stiff ODEs

Variable-step (VS) 4-stage k-step Hermite–Birkhoff (HB) methods of order p = (k + 2), p = 9, 10, denoted by HB (p), are constructed as a combination of linear k-step methods of order (p ? 2) and a diagonally implicit one-step 4-stage Runge–Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge–Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop L(a)-stable methods of order up to 11 with a > 63°. Fast algorithms are developed for solving these systems in O (p²) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsizes of these methods are controlled by a local error estimator. HB(p) of order p = 9 and 10 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7-8) and Ebadi et al. hybrid backward differentiation formulae of order 10 and 12, HBDF(10-12) in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.     

Hadamard variational formulas for <Emphasis Type="Italic">p</Emphasis>-torsion and <Emphasis Type="Italic">p</Emphasis>-eigenvalue with applications

In this paper, we pose two kinds of Minkowski problems involving the p-Laplacian operator. The Hadamard variational formulas for some p-Laplacian functionals are obtained. A good application is to prove symmetry results for solutions to some overdetermined problems of p-Laplacian equations.     

Equivalence of two definitions of a generalized <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> solution to the initial-boundary value problem for the wave equation

In our previous papers, we introduced the notion of a generalized solution to the initial-boundary value problem for the wave equation with a boundary function µ(t) such that the integral ∫ ₀ ^T (T ? t)|µ(t)| ^p dt exists. Here we prove that this solution is a unique solution to the problem in L _p that satisfies the corresponding integral identity.     

Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p)

In this paper, the parametric matrix equation A(p)X = B(p) whose elements are linear functions of uncertain parameters varying within intervals are considered. In this matrix equation A(p) and B(p) are known m-by-m and m-by-n matrices respectively, and X is the m-by-n unknown matrix. We discuss the so-called AE-solution sets for such systems and give some analytical characterizations for the AE-solution sets and a sufficient condition under which these solution sets are bounded. We then propose a modification of Krawczyk operator for parametric systems which causes reduction of the computational complexity of obtaining an outer estimation for the parametric united solution set, considerably. Then we give a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for enclosing the parametric united solution set which also enables us to reduce the computational complexity, significantly. Also some numerical approaches based on Gaussian elimination and Gauss-Seidel methods to find outer estimations for the parametric united solution set are given. Finally, some numerical experiments are given to illustrate the performance of the proposed methods.     

Positive solutions of <Emphasis Type="Italic">p</Emphasis>-th Yamabe type equations on graphs

Let G = (V,E) be a finite connected weighted graph, and assume 1 ? α ? p ? q. In this paper, we consider the p-th Yamabe type equation ―?_pu+hu^q―1 = λfu^α―1 on G, where ?_p is the p-th discrete graph Laplacian, h < 0 and f > 0 are real functions defined on all vertices of G. Instead of H. Ge’s approach [Proc. Amer. Math. Soc., 2018, 146(5): 2219–2224], we adopt a new approach, and prove that the above equation always has a positive solution u > 0 for some constant λ ∈ ?. In particular, when q = p, our result generalizes Ge’s main theorem from the case of α ? p > 1 to the case of 1 ? α ? p, It is interesting that our new approach can also work in the case of α ? p > 1.     

Approximation algorithms for <Emphasis Type="Italic">k</Emphasis>-echelon extensions of the one warehouse multi-retailer problem

In this paper, we consider k-echelon extensions of the deterministic one warehouse multi-retailer problem. We give constant factor approximation algorithms for some of these extensions when k is fixed. We focus first on the case without backorders and we give a \((2k-1)\)-approximation algorithm under general assumptions on the evolution of the holding costs as products move toward the final customers. We then improve this result to a k-approximation when the holding costs are monotonically non-increasing or non-decreasing (which is a natural situation in practice). Finally we address problems with backorders: we give a 3-approximation for the one-warehouse multi-retailer problem with backlog and a k-approximation algorithm for the k-level Joint Replenishment Problem with backlog (a variant where inventory can only be kept at the final retailers). Ours results are the first constant approximation algorithms for those problems. In addition, we demonstrate the potential of our approach on a practical case. Our preliminary experiments show that the average optimality gap is around 15%.     

Splitting theorems for pro-<Emphasis Type="Italic">p</Emphasis> groups acting on pro-<Emphasis Type="Italic">p</Emphasis> trees

Let G be an infinite finitely generated pro-p group acting on a pro-p tree such that the restriction of the action to some open subgroup is free. We prove that G splits over an edge stabilizer either as an amalgamated free pro-p product or as a pro-p \({\text {HNN}}\)-extension. Using this result, we prove under a certain condition that free pro-p products with procyclic amalgamation inherit from its amalgamated free factors the property of each 2-generated pro-p subgroup being free pro-p. This generalizes known pro-p results, as well as some pro-p analogues of classical results in abstract combinatorial group theory.