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.     

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.     

An Algorithm for Solving Capacitated Multicommodity <Emphasis Type="Italic">p</Emphasis>-median Transportation Problems

An algorithm for solving a special capacitated multicommodity p-median transportation problem (CMPMTP), which arises in container terminal management, is presented. There are some algorithms to solve similar kinds of problems. The formulation here is different from the existing modelling of the p-median or some related location problems. We extend the existing work by applying a Lagrangean relaxation to the CMPMTP. In order to obtain a satisfactory solution, a heuristic branch-and-bound algorithm is designed to search for a better solution, if one is possible. A comparison is also made with different algorithms.     

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.     

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.     

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.     

Global optimality condition and fixed point continuation algorithm for non-Lipschitz ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> regularized matrix minimization

Regularized minimization problems with nonconvex, nonsmooth, even non-Lipschitz penalty functions have attracted much attention in recent years, owing to their wide applications in statistics, control, system identification and machine learning. In this paper, the non-Lipschitz ? _p (0 < p < 1) regularized matrix minimization problem is studied. A global necessary optimality condition for this non-Lipschitz optimization problem is firstly obtained, specifically, the global optimal solutions for the problem are fixed points of the so-called p-thresholding operator which is matrix-valued and set-valued. Then a fixed point iterative scheme for the non-Lipschitz model is proposed, and the convergence analysis is also addressed in detail. Moreover, some acceleration techniques are adopted to improve the performance of this algorithm. The effectiveness of the proposed p-thresholding fixed point continuation (p-FPC) algorithm is demonstrated by numerical experiments on randomly generated and real matrix completion problems.     

<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.     

Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators

The majority of first-order methods for large-scale convex–concave saddle point problems and variational inequalities with monotone operators are proximal algorithms. To make such an algorithm practical, the problem’s domain should be proximal-friendly—admit a strongly convex function with easy to minimize linear perturbations. As a by-product, this domain admits a computationally cheap linear minimization oracle (LMO) capable to minimize linear forms. There are, however, important situations where a cheap LMO indeed is available, but the problem domain is not proximal-friendly, which motivates search for algorithms based solely on LMO. For smooth convex minimization, there exists a classical algorithm using LMO—conditional gradient. In contrast, known to us similar techniques for other problems with convex structure (nonsmooth convex minimization, convex–concave saddle point problems, even as simple as bilinear ones, and variational inequalities with monotone operators, even as simple as affine) are quite recent and utilize common approach based on Fenchel-type representations of the associated objectives/vector fields. The goal of this paper was to develop alternative (and seemingly much simpler) decomposition techniques based on LMO for bilinear saddle point problems and for variational inequalities with affine monotone operators.     

On merit functions for <Emphasis Type="Italic">p</Emphasis>-order cone complementarity problem

Merit function approach is a popular method to deal with complementarity problems, in which the complementarity problem is recast as an unconstrained minimization via merit function or complementarity function. In this paper, for the complementarity problem associated with p-order cone, which is a type of nonsymmetric cone complementarity problem, we show the readers how to construct merit functions for solving p-order cone complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also assert that these merit functions provide an error bound for the p-order cone complementarity problem. These results build up a theoretical basis for the merit method for solving p-order cone complementarity problem.     

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.     

On the complexity of inverse convex ordered 1-median problem on the plane and on tree networks

An ordered median function is used in location theory to generalize a class of problems, including median and center problems. In this paper we consider the complexity of inverse ordered 1-median problems on the plane and on trees, where the multipliers are sorted nondecreasingly. Based on the convexity of the objective function, we prove that the problems with variable weights or variable coordinates on the line are NP-hard. Then we can directly get the NP-hardness result for the corresponding problem on the plane. We finally develop a cubic time algorithm that solves the inverse convex ordered 1-median problem on trees with relaxation on modification bounds.     

An approximate solution algorithm for the one-dimensional online median problem

The online median problem consists in finding a sequence of incremental solutions of the k-median problem with k increasing. A particular case of the problem is considered: the clients and facilities are located on the real line. The best algorithm available for the one-dimensional case has competitive ratio 8. We give an improved 5.83-competitive algorithm.     

Solution of a Ball–Murat problem

A necessary and sufficient condition for the W ^{1, p} -quasi-convexity of integrands to imply the lower semicontinuity of the corresponding integral functionals with respect to the weak convergence of sequences in W ^{1, p} is obtained. It is shown that the absence of the Lavrent’ev phenomenon in minimization problems with linear boundary data is sufficient, under a minor technical assumption, for the lower semicontinuity of integral functionals with quasi-convex integrands.     

Graph clustering with a constraint on cluster sizes

A graph clustering problem is under study (also known as the graph approximation problem) with a constraint on cluster sizes. Some new approximation algorithm is presented for this problem, and performance guarantee of the algorithm is obtained. It is shown that the problem belongs to the class APX for every fixed p, where p is the upper bound on the cluster sizes.     

2-Recognizability by prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and let Γ(G) be the prime graph of G. Assume p prime. We determine the finite groups G such that Γ(G) = Γ(PSL(2, p ²)) and prove that if p ≠ 2, 3, 7 is a prime then k(Γ(PSL(2, p ²))) = 2. We infer that if G is a finite group satisfying |G| = |PSL(2, p ²)| and Γ(G) = Γ(PSL(2, p ²)) then G ? PSL(2, p ²). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications are also considered of this result to the problem of recognition of finite groups by element orders.     

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.     

An approximation algorithm for finding a <Emphasis Type="Italic">d</Emphasis>-regular spanning connected subgraph of maximum weight in a complete graph with random weights of edges

An approximation algorithm is suggested for the problem of finding a d-regular spanning connected subgraph of maximum weight in a complete undirected weighted n-vertex graph. Probabilistic analysis of the algorithm is carried out for the problem with random input data (some weights of edges) in the case of a uniform distribution of the weights of edges and in the case of a minorized type distribution. It is shown that the algorithm finds an asymptotically optimal solution with time complexity O(n ²) when d = o(n). For the minimization version of the problem, an additional restriction on the dispersion of weights of the graph edges is added to the condition of the asymptotical optimality of the modified algorithm.     

To the theory of the Dirichlet and Neumann problems for strongly elliptic systems in Lipschitz domains

For strongly elliptic Systems with Douglis-Nirenberg structure, we investigate the regularity of variational solutions to the Dirichlet and Neumann problems in a bounded Lipschitz domain. The solutions of the problems with homogeneous boundary conditions are originally defined in the simplest L ₂-Sobolev spaces H ^σ . The regularity results are obtained in the potential spaces H _p ^σ and Besov spaces B _p ^σ . In the case of second-order Systems, the author’s results obtained a year ago are strengthened. The Dirichlet problem with nonhomogeneous boundary conditions is considered with the use of Whitney arrays.