Concave Problem Research Articles

Grid-based hexahedral element meshing algorithms for solid models with concave curved boundary lines

This paper presented a grid-based hexahedral element mesh generation algorithm for solid models with concave curved boundary lines. A deep study was focused on the boundary matching and quality improvement techniques. Firstly, a method for computing the curvature values of the triangle facets and sub-surfaces was proposed. In order to improve the surface mesh quality, a layer of new elements was inserted on the surface of the jagged core mesh. Then, a relative position relationship method was used to match C-edges of the solid model. Eight different types of free quadrilateral facet configurations were established. In order to handle the concave curve-matching problem, this paper proposed a method to modify the matching properties of the degenerate quadrilateral facets fitted on the same concave curved boundary line by unifying their orientations to point to the same sub-surface. In addition, six mixed templates were newly proposed to improve the geometrical topology of the degenerate elements associated with concave curves and sharp features. The positions of the nodes were smoothed by the modified Laplacian method and objective function. Finally, the effectiveness and reliability of the algorithms proposed in this paper were demonstrated by a practical example.

Rate allocation based on spectrum pricing function in collaborative transmission over heterogeneous wireless access networks

This article studied rate allocation in collaborative transmission over heterogeneous wireless access networks. With the introduced pricing function, which uses the occupied spectrum to describe the cost in data transmission, rate allocation is formulated as a concave optimization problem and the explicit solution has been obtained. Instead of transmitting through all available networks in pursuit of fairness, the proposed rate allocation scheme distributes traffic to available networks in an unbalanced way according to both spectral efficiency and network status. Simulation results have shown that compared with other methods, the new scheme can maximize the total utility gained in collaborative transmission and avoid congestion effectively.

Fault-tolerant concave facility location problem with uniform requirements

In this paper, we consider the fault-tolerant concave facility location problem (FTCFL) with uniform requirements. By investigating the structure of the FTCFL, we obtain a modified dual-fitting bifactor approximation algorithm. Combining the scaling and greedy argumentation technique, the approximation factor is proved to be 1.52.

Regularizability of Ill-Posed Problems and the Modulus of Continuity

The regularization of linear ill-posed problems is based on their conditional well-posedness when restricting the problem to certain classes of solutions. Given such class one may consider several related real-valued functions, which measure the well-posedness of the problem on such class. Among those functions the modulus of continuity is best studied. For solution classes which enjoy the additional feature of being star-shaped at zero, the authors develop a series of results with focus on continuity properties of the modulus of continuity. In particular it is highlighted that the problem is conditionally well-posed if and only if the modulus of continuity is right-continuous at zero. Those results are then applied to smoothness classes in Hilbert space. This study concludes with a new perspective on a concavity problem for the modulus of continuity, recently addressed by two of the authors in [Trudy Inst. Mat. i Mekh. UrO RAN 18 (2012)(1), 34{41].

Global optimal solutions to a class of quadrinomial minimization problems with one quadratic constraint

This paper studies the canonical duality theory for solving a class of quadrinomial minimization problems subject to one general quadratic constraint. It is shown that the nonconvex primal problem in $${\mathbb {R}^n}$$ can be converted into a concave maximization dual problem over a convex set in $${\mathbb {R}^2}$$ , such that the problem can be solved more efficiently. The existence and uniqueness theorems of global minimizers are provided using the triality theory. Examples are given to illustrate the results obtained.

Joint subcarrier and power allocation for uplink relay-enhanced OFDM systems

There has been a lot of research works considering the resource allocation of the downlink multihop orthogonal frequency division multiplexing systems. However, due to the distributed nature of the uplink power constraints, the resource allocation in the uplink multihop systems, where multiple mobile stations transmit to one base station with the aid of one or many relay stations, has much difference and has not been well investigated so far. In this paper, we originally study the joint subcarrier and power allocation problem for the uplink dual-hop transmission with the aim to maximize the system transmit rate. The resource allocation problem is approximated to be a concave maximization problem. By using mathematical decomposition techniques, the problem is first decoupled and solved by the proposed near-optimal method, which has low-computation complexity. Then, our algorithm is extended to the case with subcarrier matching on the dual hops. Numerical results show that our proposed algorithm improves the system transmission rate. Compared with the equal power allocation schemes, our algorithm can achieve significant gain in system transmit rate. Copyright © 2010 John Wiley & Sons, Ltd.

Optimality conditions in the problem of maximization of the difference of two convex functions

We consider a quadratic d. c. optimization problem on a convex set. The objective function is represented as the difference of two convex functions. By reducing the problem to the equivalent concave programming problem we prove a sufficient optimality condition in the form of an inequality for the directional derivative of the objective function at admissible points of the corresponding level surface.

Unboundedness in reverse convex and concave integer programming

In this paper we are concerned with the problem of unboundedness and existence of an optimal solution in reverse convex and concave integer optimization problems. In particular, we present necessary and sufficient conditions for existence of an upper bound for a convex objective function defined over the feasible region contained in \({\mathbb{Z}^n}\). The conditions for boundedness are provided in a form of an implementable algorithm, showing that for the considered class of functions, the integer programming problem is unbounded if and only if the associated continuous problem is unbounded. We also address the problem of boundedness in the global optimization problem of maximizing a convex function over a set of integers contained in a convex and unbounded region. It is shown in the paper that in both types of integer programming problems, the objective function is either unbounded from above, or it attains its maximum at a feasible integer point.

A particle swarm optimization-based hybrid algorithm for minimum concave cost network flow problems

Traditionally, minimum cost transshipment problems have been simplified as linear cost problems, which are not practical in real applications. Some advanced local search algorithms have been developed to solve concave cost bipartite network problems. These have been found to be more effective than the traditional linear approximation methods and local search methods. Recently, a genetic algorithm and an ant colony system algorithm were employed to develop two global search algorithms for solving concave cost transshipment problems. These two global search algorithms were found to be more effective than the advanced local search algorithms for solving concave cost transshipment problems. Although the particle swarm optimization algorithm has been used to obtain good results in many applications, to the best of our knowledge, it has not yet been applied in minimum concave cost network flow problems. Thus, in this study, we employ an arc-based particle swarm optimization algorithm, coupled with some genetic algorithm and threshold accepting method techniques, as well as concave cost network heuristics, to develop a hybrid global search algorithm for efficiently solving minimum cost network flow problems with concave arc costs. The proposed algorithm is evaluated by solving several randomly generated network flow problems. The results indicate that the proposed algorithm is more effective than several other recently designed methods, such as local search algorithms, genetic algorithms and ant colony system algorithms, for solving minimum cost network flow problems with concave arc costs.

Sign reversion approach to concave minimization problems

We consider a problem of minimization of a concave function subject to affine constraints. By using sign reversion techniques we show that the initial problem can be transformed into a family of concave maximization problems. This property enables us to suggest certain algorithms based on the parametric dual optimization problem.

Implementation of FIR control for H ∞ output feedback stabilisation of linear systems

In this article, finite impulse response (FIR) control is addressed for H ∞ output feedback stabilisation of linear systems. The problem we deal with is the construction of an output feedback controller with a certain FIR structure such that the resulting closed-loop system is asymptotically stable and a prescribed H ∞ norm bound constraint is guaranteed. Some solvability conditions are suggested in this article. Sufficient conditions are derived to obtain a suboptimal solution of the H ∞ FIR control problem via convex optimisation. Also, an equivalent condition for the existence of H ∞ FIR control is presented in the set of linear matrix inequalities (LMIs) and a reciprocal matrices equality constraint. An effective computational algorithm involving LMIs is suggested to solve a concave minimisation problem characterising a local optimal solution of the H ∞ FIR control problem. Numerical examples demonstrate the validity of the proposed H ∞ FIR control and the numerical efficiency of the proposed algorithm for FIR control.

Regularity for obstacle problems in infinite dimensional Hilbert spaces

In this paper we study fully nonlinear obstacle-type problems in Hilbert spaces. We introduce the notion of Q-elliptic equation and prove existence, uniqueness, and regularity of viscosity solutions of Q-elliptic obstacle problems. In particular we show that solutions of concave problems with semiconvex obstacles are in the space WQ2,∞.

Solution to nonconvex quadratic programming with both inequality and box constraints

This paper presents a canonical dual approach for solving nonconvex quadratic programming problems subjected to both linear inequality constraints and box constrains. It is proved that the constrained nonconvex primal problem can be reformulated as a concave maximization dual problem with zero duality gap, which can be solved under certain conditions. Both global and local extremimality conditions are presented by the triality theory. Several applications are illustrated.

An adaptive tabu-simulated annealing for concave cost transportation problems

The transportation problem (TP) is one of the most popular network problems because of its theoretical and practical importance. If the transportation cost linearly depends on the transported amount of the product, then TP is solvable in polynomial time with linear programming methods. However, in the real world, the transportation costs are generally nonlinear, frequently concave where the unit cost for transporting products decreases as the amount of products increases. Since concave cost transportation problems (ccTPs) are NP-hard, solving large-scale problems is time consuming. In this study, we propose a hybrid algorithm based on the concepts borrowed from tabu search (TS) and simulated annealing (SA) to solve the ccTP. This algorithm, called ATSA (adaptive tabu-simulated annealing), is an SA approach supplemented with a tabu list and adaptive cooling strategy. The effectiveness of ATSA has been investigated in two stages using a set of TPs with different sizes. The first stage includes performance analysis of ATSA using SA, ASA (adaptive simulated anealing) and TS, which are basic forms of ATSA. In the second stage, ATSA has been compared with the heuristic approaches given in the literature for ccTP. Statistical analysis shows that ATSA exhibits better performance than its basic forms and heuristic approaches.

A Path Following Algorithm for the Graph Matching Problem

We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We, therefore, construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore, perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four data sets: simulated graphs, QAPLib, retina vessel images, and handwritten Chinese characters. In all cases, the results are competitive with the state of the art.

A generalized projective dynamic for solving extreme and interior eigenvalue problems

In [18] (Golub and Liao), a continuous-time system which is based on the projective dynamic is proposed to solve some concave optimization problems (with the unit ball constraint) resulted from extreme and interior eigenvalue problems. The convergence inside the unit ball is established; however, neither further convergence result outside the unit ball nor the stability analysis is available. Moreover, preliminary numerical experience indicates that this method is sensitive to a parameter whose optimal value is still difficult to determine. After analyzing the stability of this dynamic, in this paper, we develop a generalized model and analyze the convergence of the new model both inside and outside the unit ball. The flow of the generalized model is proved to converge almost globally to some eigenvector corresponding to the smallest eigenvalue, and share many surprisingly analogous properties with the Rayleigh quotient gradient flow. Links of our generalized projective dynamical system with other related works are also discussed. The efficiency of our new model is both addressed in theory and verified in numerical testing.

On some general almost periodic Optimal Control problems: links with periodic problems and necessary conditions

In this paper, we are concerned with periodic, quasi-periodic (q.p.) and almost periodic (a.p.) Optimal Control problems. After defining these problems and setting them in an abstract setting by using Abstract Harmonic Analysis, we give some structure results of the set of solutions, and study the relations between periodic and a.p. problems. We prove for instance that for an autonomous concave problem, the a.p. problem has a solution if and only if all problems (periodic with fixed or variable period, q.p. or a.p.) have a constant solution. After that, we give some first order necessary conditions (weak Pontryagin) in the space of Harmonic Synthesis and we also give in this space an existence result.

Concave programming and DH-point

An extreme point property of optimal solutions of general concave programming problems is established that generalizes both Du-Hwang's minimax theorem and its continuous version by Du and Pardalos.

A linearized finite-difference method for the solution of some mixed concave and convex non-linear problems

In the present work, a linearized finite-difference scheme is proposed in order to approximate the solution of some non-linear equation characterized by mixed concave and convex non-linearities. It is proved that the scheme is uniquely solvable and convergent. The resulting method is also analyzed for consistency and stability. Some numerical examples illustrating the method described in our work are given.

An Efficient Algorithm for Capacitated Multifacility Location Problems

In this paper, a squared-Euclidean distance multifacility location problem with inseparable demands under balanced transportation constraints is analyzed. Using calculus to project the problem onto the space of allocation variables, the problem becomes minimizing concave quadratic integer programming problem. The algorithm based on extreme point ranking method combining with logical techniques is developed. The numerical experiments are randomly generated to test efficiency of the proposed algorithm compared with a linearization algorithm. The results show that the proposed algorithm provides a better solution on average with less processing time for all various sizes of problems.