Concave Programming Problem Research Articles

Energy efficient resource allocation in cache‐enabled fog networks

AbstractTo gain high energy efficiency and low latency, fog computing can be considered as a promising enabling technology for supporting future IoT networks. Exponentially increasing data rate demand of smart IoT devices in the fifth‐Generation (5G) networks require new resource management and power allocation schemes. Fog computing involves the deployment of data storage devices (fog nodes) near the proximity of IoT nodes. The fog nodes collaborate to process requests generated from users. These nodes can store data in their storage capacity and supports in achieving quality‐of‐service (QoS) requirements. A joint node association and energy efficiency maximization problem is formulated in this article. The proposed problem is a non‐linear concave programming problem under cache size, association, and optimal power allocation constraints. A mesh adaptive direct search algorithm (MADS) is used to solve the formulated problem to find the sub‐optimal results. The efficient performance of the algorithm can be seen in comparison with more complex algorithms namely, outer approximation algorithm and exhaustive search algorithm. Extensive simulations are done to observe the detailed performance analysis of the IoT‐Fog network. As a result, energy‐efficient resource allocation is done under cache and QoS constraints with very little complexity and low computational power and delays.

Energy-efficient power allocation for massive MIMO-enabled multi-way AF relay networks with channel aging

In this paper, we consider a massive MIMO-enabled multi-way amplify-and-forward relay network with channel aging, where multiple users mutually exchange information via an intermediate relay equipped with massive antennas. For this system, we propose an energy-efficient power allocation scheme for the optimization of energy efficiency (EE). Specifically, we firstly derive accurate closed-form expressions of the system sum rate with aged channel state information (CSI) and predicted CSI. Secondly, based on the derived analytical results, a unified power allocation optimization problem with aged/predicted CSI is formulated for maximizing the system EE. To solve this challenging problem, the successive convex approximation technique is invoked to transform the original optimization problem into a tractable concave fractional programming problem. Then, Dinkelbach’s algorithm and Lagrangian dual method are adopted to find the optimal solution. In addition, to strike a balance between the computational complexity and the optimality, the EE maximization problem using the equal power allocation scheme is solved by extreme value theorem, leading to a closed-form optimal solution. Numerical results demonstrate the accuracy of our analytical results and the effectiveness of the proposed algorithms. Moreover, the impact of several important system parameters on the system performance achieved by the proposed algorithms is also illustrated.

One algorithm for branch and bound method for solving concave optimization problem

The article describes the algorithm for branch and bound method for solving the concave programming problem, which is based on the idea of similarity the necessary and sufficient conditions of optimum for the original problem and for a convex programming problem with another feasible set and reverse the sign of the objective function. To find the feasible set of the equivalent convex programming problem we construct an algorithm using the idea of the branch and bound method. We formulate various branching techniques and discusses the construction of the lower objective function evaluations for the node of the decision tree. The article discusses the results of experiments of this algorithm for some famous test problems of a particular form.

Decomposition Procedure for Solving NLP and QP Problems based on Lagrange and Sander's Method

This paper develops a decompose procedure for finding the optimal solution of convex and concave Quadratic Programming (QP) problems together with general Non-linear Programming (NLP) problems. The paper also develops a sophisticated computer technique corresponding to the author's algorithm using programming language MATHEMATICA. As for auxiliary by making comparison, the author introduces a computer-oriented technique of the traditional Karush-Kuhn-Tucker (KKT) method and Lagrange method for solving NLP problems. He then modify the Sander's algorithm and develop a new computational technique to evaluate the performance of the Sander's algorithm for solving NLP problems. The author observe that the technique avoids some certain numerical difficulties in NLP and QP. He illustrates a number of numerical examples to demonstrate his method and the modified algorithm.

Analysis of a Joint Pricing and Seat Allocation Model in a Hub-to-Hub Airline Network

In airline revenue management, after differentiating products on each itinerary according to various restrictions, management needs to set the prices for the products. A deterministic joint pricing and seat allocation model is proposed. It is reduced to a separable concave programming problem and thus readily solvable. Focusing on a special hub-to-hub airline network, monotonicity of the pricing decisions is explored. Using a nonlinear primal–dual technique, it is shown that some itineraries’ optimal prices are decreasing, while some other itineraries’ optimal prices are increasing with a capacity change in either a side leg or the middle leg. Such structural properties add important managerial insights into the pricing model for revenue managers in an airline company.

CDF formulation for solving an optimal reinsurance problem

An innovative cumulative distribution function (CDF)-based method is proposed for deriving optimal reinsurance contracts to maximize an insurer’s survival probability. The optimal reinsurance model is a non-concave constrained stochastic maximization problem, and the CDF-based method transforms it into a functional concave programming problem of determining an optimal CDF over a corresponding feasible set. Compared to the existing literature, our proposed CDF formulation provides a more transparent derivation of the optimal solutions, and more interestingly, it enables us to study a further complex model with an extra background risk and more sophisticated premium principle.

Joint optimisation of buffers and network population for closed finite queueing systems

The buffer allocation problem for closed finite queueing networks is a nonlinear integer programming problem which is -Hard. A queueing network decomposition methodology based upon a mean value analysis algorithm is coupled with a nonlinear sequential quadratic programming algorithm to compute the simultaneous optimal buffer allocations and network population via a branch-and-bound scheme for various network topologies. The joint optimisation problem is essentially multi-objective where the decision variable on the population will be treated as a constraint linking the buffer allocation and the throughput which is to be maximised. The fact that the optimisation problem is a concave programming problem assists in the search for an optimal solution. The material handling or transportation system for transferring the finite customer population between the nodes in the network is also examined. Extensive numerical results demonstrate the efficacy of the optimisation methodology.

Optimization Methods for Mixed Integer Weakly Concave Programming Problems

In this paper, we consider a class of mixed integer weakly concave programming problems (MIWCPP) consisting of minimizing a difference of a quadratic function and a convex function. A new necessary global optimality conditions for MIWCPP is presented in this paper. A new local optimization method for MIWCPP is designed based on the necessary global optimality conditions, which is different from the traditional local optimization method. A global optimization method is proposed by combining some auxiliary functions and the new local optimization method. Furthermore, numerical examples are also presented to show that the proposed global optimization method for MIWCPP is efficient.

Existence of Competitive Equilibrium in Linear Exchange Economies: A Simple Proof Using Brouwer’s Fixed Point Theorem

In this paper we show that for concave piecewise linear exchange economies every competitive equilibrium satisfies the property that the competitive allocation is a non-symmetric Nash bargaining solution with weights being the initial income of individual agents evaluated at the equilibrium price vector. We prove the existence of competitive equilibrium for concave piecewise linear exchange economies by obtaining a non-symmetric Nash bargaining solution with weights being defined appropriately. In a later section we provide a simpler proof of the same result using the Brouwer’s fixed point theorem, when all utility functions are linear. In both cases the proofs pivotal step is the concave maximization problem due to Eisenberg and Gale and minor variations of it. We also provide a proof of the same results for economies where agents’ utility functions are concave, continuously differentiable and homogeneous. In this case the main argument revolves around the concave maximization problem due to Eisenberg. Unlike previous results, we do not require all initial endowments to lie on a fixed ray through the origin. 1. Introduction : Eisenberg and Gale (1959) showed that for linear exchange economies with individual initial endowment bundles lying on a fixed ray through the origin, the proof of existence of a competitive equilibrium reduces to finding a solution for a concave programming problem. In particular they showed that the corresponding competitive allocation is a nonsymmetric Nash bargaining solution (see Kalai (1977)) with weights being given by the share of each individual in the aggregate initial endowment. Moore (2007) refers to the non-symmetric Nash bargaining solution as a solution to the Cobb-Douglas-Eisenberg (CDE) aggregator function. In this paper we first show that for concave piecewise linear exchange economies every competitive equilibrium satisfies the property that a competitive allocation is a non-symmetric

Simultaneous confidence bounds in monotone dose–response studies

In dose–response studies, one of the most important issues is the identification of minimum effective dose (MED), where the MED is defined as the lowest dose such that the mean response is better than the mean response of a zero-dose control by a clinically significant difference. Dose–response curves are sometimes monotonic in nature. In this paper, we propose a new method to construct simultaneous confidence lower bounds for the differences between the mean response of any non-zero dose level and that of the control under the monotonicity assumption to identify the MED. The evaluation of the lower confidence bounds is a concave programming problem subject to homogeneous linear inequality constraints. A necessary and sufficient condition for the solution and its simplified expressions are derived. An efficient computing algorithm is proposed. A real data example from a dose–response study is used to illustrate the method.

On Box Constrained Concave Quadratic Optimization

This paper presents a class of differential flows to solve concave quadratic programming problems under box constraints. Some properties of the flow are given to reveal the significant relationship between the dynamic of the flow and the geometry of the feasible set. It is shown how the differential flow reaches a vertex point of the box, leading to a global minimizer of the concave quadratic programming. Some illustrative examples are also presented.

Optimal decision making in multi-product dual sourcing procurement with demand forecast updating

In this paper, we will investigate a buyer's decision making problem in procuring multiple products, each treated as a newsvendor, from two markets. The contract market has a long lead time, a fixed wholesale price and resource constraints. While the spot market has an instant lead time and a highly volatile price. The purchasing decision at the spot market can be made near the beginning of the selling season to take the advantage of the most recent demand forecast. The buyer needs to determine the purchasing quantity for each product at the two markets to maximize the expected profit by trading off between the resource availability, demand uncertainty and price variability. The procurement decision making is modeled as a bi-level programming problem under both a single resource constraint and under multiple resource constraints. We show that this bi-level programming problem can be formulated as a single-level concave programming problem. We then develop a sequential algorithm which solves for a linear approximation of the concave programming problem in each iteration. This algorithm can be used to solve a real world problem with up to thousands of kinds of products, and is found to be highly efficient and effective.

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.

New results on the equivalence between zero-one programming and continuous concave programming

In this work, we study continuous reformulations of zero-one concave programming problems. We introduce new concave penalty functions and we prove, using general equivalence results here derived, that the obtained continuous problems are equivalent to the original combinatorial problem.

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.

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.

Statistical Inference for the Difference Between the Best Treatment Mean and a Control Mean

In many experiments, researchers are interested in comparing several treatment means with a control mean. When there are some treatments significantly better than the control, it is often of interest to evaluate the difference between the best treatment mean and the control mean and to identify the best treatment. In this article we derive lower confidence bounds for the aforementioned difference for the case that treatments are at least as effective as the control and for the case that no restriction is placed on the treatment means and the control mean. The evaluation of the lower confidence bound for the difference between the best treatment mean and the control mean is a concave programming problem subject to homogeneous linear inequality constraints. We propose two efficient computation algorithms and discuss the connection between our procedures and Gupta's subset selection procedure. We compare the expected lower confidence bounds of the two procedures with that of Dunnett's procedure. An application to a real-life data is included.

Global minimization algorithms for concave quadratic programming problems

: In this article, we consider the concave quadratic programming problem which is known to be NP hard. Based on the improved global optimality conditions by [Dür, M., Horst, R. and Locatelli, M., 1998, Necessary and sufficient global optimality conditions for convex maximization revisited, Journal of Mathematical Analysis and Applications, 217, 637–649] and [Hiriart-Urruty, J.B. and Ledyav, J.S., 1996, A note in the characterization of the global maxima of a convex function over a convex set, Journal of Convex Analysis, 3, 55–61], we develop a new approach for solving concave quadratic programming problems. The main idea of the algorithms is to generate a sequence of local minimizers either ending at a global optimal solution or at an approximate global optimal solution within a finite number of iterations. At each iteration of the algorithms we solve a number of linear programming problems with the same constraints of the original problem. We also present the convergence properties of the proposed algorithms under some conditions. The efficiency of the algorithms has been demonstrated with some numerical examples.

Generalized linearly homogeneous production functions

The production functions are considered as functions of values of concave programming problems. Some properties of the functions, in particular, the property of homogeneity and linear homogeneity, are investigated. Examples of the construction of such functions are presented.

An Outer Approximation Algorithm Guaranteeing Feasibility of Solutions and Approximate Accuracy of Optimality

We treat a concave programming problem with a compact convex feasible set. Assuming the differentiability of the convex functions which define the feasible set, we propose two solution methods. Those methods utilize the convexity of the feasible set and the property of the normal cone to the feasible set at each point over the boundary. Based on the proposed two methods, we propose a solution algorithm. This algorithm takes advantages over classical methods: (1) the obtained approximate solution is always feasible, (2) the error of such approximate value can be evaluated properly for the optimal value of such problem, (3) the algorithm does not have any redundant iterations.