Variational Analysis on Local Sharp Minima via Exact Penalization

In this paper we study local sharp minima of the nonlinear programming problem via exact penalization. Utilizing generalized differentiation tools in variational analysis such as subderivatives and regular subdifferentials, we obtain some primal and dual characterizations for a penalty function associated with the nonlinear programming problem to have a local sharp minimum. These general results are then applied to the ? _p penalty function with 0 ≤ p ≤ 1. In particular, we present primal and dual equivalent conditions in terms of the original data of the nonlinear programming problem, which guarantee that the ? _p penalty function has a local sharp minimum with a finite penalty parameter in the case of \(p\in (\frac {1}{2}, 1]\) and \(p=\frac {1}{2}\) respectively. By assuming the Guignard constraint qualification (resp. the generalized Guignard constraint qualification), we also show that a local sharp minimum of the nonlinear programming problem can be an exact local sharp minimum of the ? _p penalty function with p ∈ [0, 1] (resp. \(p\in [0, \frac {1}{2}]\)). Finally, we give some formulas for calculating the smallest penalty parameter for a penalty function to have a local sharp minimum.     

Minimization of transformed $$L_1$$ penalty: theory,difference of convex function algorithm,and robust application in compressed sensing

We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed \(l_1\) (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates \(l_0\) and \(l_1\) norms through a nonnegative parameter \(a \in (0,+\infty )\), similar to \(l_p\) with \(p \in (0,1]\), and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of \(l_0\) norm minimal solution based on the null space property (NSP). We then prove the stable recovery of \(l_0\) norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an \(l_1\) minimization problem on which we employ the Alternating Direction Method of Multipliers. For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value \(a=1\), and compare DCATL1 with other CS algorithms on two classes of sensing matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on \(l_{1/2}\) norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on \(l_1\) minus \(l_2\) penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with \(l_1\) minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.     

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.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

The exact absolute value penalty function method for identifying strict global minima of order <Emphasis Type="Italic">m</Emphasis> in nonconvex nonsmooth programming

In this paper, it is demonstrated that the exact absolute value penalty function method is useful for identifying the special sort of minimizers in nonconvex nonsmooth optimization problems with both inequality and equality constraints. The equivalence between the sets of strict global minima of order m in nonsmooth minimization problem and of its associated penalized optimization problem with the exact \(l_{1}\) penalty function is established under nondifferentiable \(\left( F,\rho \right) \)-convexity assumptions imposed on the involved functions. The threshold of the penalty parameter, above which this result holds, is also given.     

On teaching sets for 2-threshold functions of two variables

We consider k-threshold functions of n variables, i.e. the functions representable as the conjunction of k threshold functions. For n = 2, k = 2, we give upper bounds for the cardinality of the minimal teaching set depending on the various properties of the function.     

Domination Defect in Graphs: Guarding With Fewer Guards

In this paper, we introduce a new graph parameter called the domination defect of a graph. The domination number γ of a graph G is the minimum number of vertices required to dominate the vertices of G. Due to the minimality of γ, if a set of vertices of G has cardinality less than γ then there are vertices of G that are not dominated by that set. The k-domination defect of G is the minimum number of vertices which are left un-dominated by a subset of γ - k vertices of G. We study different bounds on the k-domination defect of a graph G with respect to the domination number, order, degree sequence, graph homomorphisms and the existence of efficient dominating sets. We also characterize the graphs whose domination defect is 1 and find exact values of the domination defect for some particular classes of graphs.     

New and efficient algorithms for transfer prices and inventory holding policies in two-enterprise supply chains

We consider a multi-period problem of fair transfer prices and inventory holding policies in two enterprise supply chains. This problem was formulated as a mixed integer non-linear program by Gjerdrum et al. (Eur J Oper Res 143:582–599, 2002). Existing global optimization methods to solve this problem are computationally expensive. We propose a continuous approach based on difference of convex functions (DC) programming and DC Algorithms (DCA) for solving this combinatorial optimization problem. The problem is first reformulated as a DC program via an exact penalty technique. Afterward, DCA, an efficient local approach in non-convex programming framework, is investigated to solve the resulting problem. For globally solving this problem, we investigate a combined DCA-Branch and Bound algorithm. DCA is applied to get lower bounds while upper bounds are computed from a relaxation problem. The numerical results on several test problems show that the proposed algorithms are efficient: DCA provides a good integer solution in a short CPU time although it works on a continuous domain, and the combined DCA-Branch and Bound finds an \(\epsilon \) -optimal solution for large-scale problems in a reasonable time.     

Maximal subsets free of arithmetic progressions in arbitrary sets

The problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length k in a given set of size n is considered. It is proved that it is sufficient, in a certain sense, to consider the interval [1,..., n]. The study continues the work of Komlós, Sulyok, and Szemerédi.     

Conditional connectivity of bubble sort graphs

A subset F ? V (G) is called an R ^k -vertex-cut of a graph G if G ? F is disconnected and each vertex of G ? F has at least k neighbors in G ? F. The R ^k -vertex-connectivity of G, denoted by κ ^k (G), is the cardinality of a minimum R ^k -vertex-cut of G. Let B _n be the bubble sort graph of dimension n. It is known that κ _k (B _n ) = 2 ^k (n ? k ? 1) for n ≥ 2k and k = 1, 2. In this paper, we prove it for k = 3 and conjecture that it is true for all k ∈ N. We also prove that the connectivity cannot be more than conjectured.     

2-Domination number of generalized Petersen graphs

Let \(G=(V,E)\) be a graph. A subset \(S\subseteq V\) is a k-dominating set of G if each vertex in \(V-S\) is adjacent to at least k vertices in S. The k-domination number of G is the cardinality of the smallest k-dominating set of G. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs \(P(5k+1, 2)\) and \(P(5k+2, 2)\), for \(k>0\), is \(4k+2\) and \(4k+3\), respectively. This proves two conjectures due to Cheng (Ph.D. thesis, National Chiao Tung University, 2013). Moreover, we determine the exact 2-domination number of generalized Petersen graphs P(2k, k) and \(P(5k+4,3)\). Furthermore, we give a good lower and upper bounds on the 2-domination number of generalized Petersen graphs \(P(5k+1, 3), P(5k+2,3)\) and \(P(5k+3, 3).\)     

The Local Metric Dimension of Strong Product Graphs

A vertex \(v\in V(G)\) is said to distinguish two vertices \(x,y\in V(G)\) of a nontrivial connected graph G if the distance from v to x is different from the distance from v to y. A set \(S\subset V(G)\) is a local metric generator for G if every two adjacent vertices of G are distinguished by some vertex of S. A local metric generator with the minimum cardinality is called a local metric basis for G and its cardinality, the local metric dimension of G. It is known that the problem of computing the local metric dimension of a graph is NP-Complete. In this paper we study the problem of finding exact values or bounds for the local metric dimension of strong product of graphs.     

An approximation scheme for a problem of search for a vector subset

We consider the following clustering problem: Given a vector set, find a subset of cardinality k and minimum square deviation from its mean. The distance between the vectors is defined by the Euclideanmetric. We present an approximation scheme (PTAS) that allows us to solve this problem with an arbitrary relative error ? in time O(n ^2/?+1(9/?)^3/? d), where n is the number of vectors of the input set and d denotes the dimension of the space.     

The Upper Connected Vertex Detour Monophonic Number of a Graph

For any vertex x in a connected graph G of order n ≥ 2, a set S _x ? V (G) is an x-detour monophonic set of G if each vertex v ∈ V (G) lies on an x-y detour monophonic path for some element y in S _x . The minimum cardinality of an x-detour monophonic set of G is the x-detour monophonic number of G, denoted by dm _x (G). A connected x-detour monophonic set of G is an x-detour monophonic set S _x such that the subgraph induced by S _x is connected. The minimum cardinality of a connected x-detour monophonic set of G is the connected x-detour monophonic number of G, denoted by cdm _x (G). A connected x-detour monophonic set S _x of G is called a minimal connected x-detour monophonic set if no proper subset of S _x is a connected x-detour monophonic set. The upper connected x-detour monophonic number of G, denoted by cdm⁺ _x (G), is defined to be the maximum cardinality of a minimal connected x-detour monophonic set of G. We determine bounds and exact values of these parameters for some special classes of graphs. We also prove that for positive integers r,d and k with 2 ≤ r ≤ d and k ≥ 2, there exists a connected graph G with monophonic radius r, monophonic diameter d and upper connected x-detour monophonic number k for some vertex x in G. Also, it is shown that for positive integers j,k,l and n with 2 ≤ j ≤ k ≤ l ≤ n - 3, there exists a connected graph G of order n with dm _x (G) = j,dm⁺ _x (G) = k and cdm⁺ _x (G) = l for some vertex x in G.     

An improved approximation algorithm for the <Emphasis Type="Italic">k</Emphasis>-level facility location problem with soft capacities

We consider the k-level facility location problem with soft capacities (k-LFLPSC). In the k-LFLPSC, each facility i has a soft capacity u _i along with an initial opening cost f _i ≥ 0, i.e., the capacity of facility i is an integer multiple of u _i incurring a cost equals to the corresponding multiple of f _i . We firstly propose a new bifactor (ln(1/β)/(1 ?β),1+2/(1 ?β))-approximation algorithm for the k-level facility location problem (k-LFLP), where β ∈ (0, 1) is a fixed constant. Then, we give a reduction from the k-LFLPSC to the k-LFLP. The reduction together with the above bifactor approximation algorithm for the k-LFLP imply a 5.5053-approximation algorithm for the k-LFLPSC which improves the previous 6-approximation.     

The problem of a minimal ball enclosing <Emphasis Type="Italic">k</Emphasis> points

Under study is the problem of finding a ball of minimal radius enclosing at least k points of a given finite set in a Euclidean space. In the case of a fixed dimension of the space this problem is polynomially solvable, but in general its complexity has not been previously determined. We prove that the problem is NP-hard in the strong sense and obtain a polynomial-time approximation scheme (PTAS) that enables us to solve the problem with an arbitrary relative error ? in time \(O(n^{1/\varepsilon ^2 + 1} d)\), where n is the cardinality of the original set and d is the space dimension.     

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 Uncountable Lattice of All Subfunctors of the Functor of Rational Representations

In set theory the cardinality of the continuum \(|{\mathbb R}|\) is the cardinal number of some interesting sets, like the Cantor set or the transcendental numbers. We will prove that the cardinal number of all subfunctors of the functor of rational representations \(k \otimes_{{\mathbb Z}} R_{{\mathbb Q}}\), taking values on odd order groups over the field k of characteristic 2, is equal to \(|{\mathbb R}|\). When the characteristic q?>?0 of the field k is not necessarily even, we will present a formula giving the dimension of the evaluations S _C,k(G), of the simple functor S _C,k, at any group G of order prime to q and being associated to a suitable cyclic group C.     

On the minimum-cost $$\lambda $$-edge-connected k-subgraph problem

In this paper, we propose several integer programming (IP) formulations to exactly solve the minimum-cost \(\lambda \)-edge-connected k-subgraph problem, or the \((k,\lambda )\)-subgraph problem, based on its graph properties. Special cases of this problem include the well-known k-minimum spanning tree problem (if \(\lambda =1\)), \(\lambda \)-edge-connected spanning subgraph problem (if \(k=|V|\)) and k-clique problem (if \(\lambda = k-1\) and there are exact k vertices in the subgraph). As a generalization of k-minimum spanning tree and a case of the \((k,\lambda )\)-subgraph problem, the (k, 2)-subgraph problem is studied, and some special graph properties are proved to find stronger and more compact IP formulations. Additionally, we study the valid inequalities for these IP formulations. Numerical experiments are performed to compare proposed IP formulations and inequalities.     

Heuristic algorithms for general <Emphasis Type="Italic">k</Emphasis>-level facility location problems

In a general k-level uncapacitated facility location problem (k-GLUFLP), we are given a set of demand points, denoted by D, where clients are located. Facilities have to be located at a given set of potential sites, which is denoted by F in order to serve the clients. Each client needs to be served by a chain of k different facilities. The problem is to determine some sites of F to be set up and to find an assignment of each client to a chain of k facilities so that the sum of the setup costs and the shipping costs is minimized. In this paper, for a fixed k, an approximation algorithm within a factor of 3 of the optimum cost is presented for k-GLUFLP under the assumption that the shipping costs satisfy the properties of metric space. In addition, when no fixed cost is charged for setting up the facilities and k=2, we show that the problem is strong NP-complete and the constant approximation factor is further sharpen to be 3/2 by a simple algorithm. Furthermore, it is shown that this ratio analysis is tight.