Separable Programming Problems Research Articles

Separable programming problem with bipolar max-T fuzzy relation equation constraints

One of the most important classes of nonlinear programming problems is separable programming problem due to its applications in theory and practice. This paper studies this class of the problems subject to a system of bipolar fuzzy relation equations using the max-T composition operator, where T is a continuous and Archimedean t-norm. Its feasible solution set structure is determined by two vectors called lower and upper bound vector of its feasible domain. It is shown that there exists an optimal solution for the problem with max-continuous and Archimedean t-norm such that its components are the corresponding components of either the lower or upper bound vector. Based on the interesting property, some sufficient conditions are proposed to detect some of its optimal components or one of its optimal solutions. These conditions can reduce the dimensions of the original problem when some its optimal components are determined. The objective function of the reduced problem is equivalently rewritten in a specific form. This form guides us to design a value matrix based on the characteristic matrix and ascending or descending of univariate functions constituting the objective function. An approach is extended on the matrix to find the optimal solution of the (reduced) problem. It is important to note that a number of problems can completely be solved using the sufficient conditions.

Numerical solution of box constrained separable convex quadratic programming problems

In this paper, minimization problems with a separable convex quadratic objective function subject to a linear equality constraint/linear equality constraints, and bounds on the variables (box constraints) are considered. A necessary and sufficient condition for a feasible solution to be an optimal solution to each of these problems has been established. A convergent polynomial algorithm for solving the problem with a linear equality constraint and bounded variables is proposed, and some numerical results, obtained by this algorithm, are presented.

An optimization method to solve a fully intuitionistic fuzzy non-linear separable programming problem

This paper presents an optimization method to solve a non-linear separable programming problem with coefficients and variables as generalized trapezoidal intuitionistic fuzzy numbers. Such optimization problems are known as fully intuitionistic fuzzy non-linear separable programming problems. The optimization method is based on the linear approximation of fully intuitionistic fuzzy non-linear separable functions. The concept of an intuitionistic fuzzy line segment between two intuitionistic fuzzy points is introduced to find the required linear approximation. In this way, a fully intuitionistic fuzzy non-linear programming problem is converted into an intuitionistic fuzzy linear programming problem. The defuzzification and component-wise comparison techniques are then used to convert the fully intuitionistic fuzzy linear programming problem to a linear programming problem with crisp coefficients which can then be solved by using traditional optimization techniques. The application of the proposed approach in an investment problem faced by a businessman has been presented.

A rank-two relaxed parallel splitting version of the augmented Lagrangian method with step size in (0,2) for separable convex programming

The augmented Lagrangian method (ALM) is classic for canonical convex programming problems with linear constraints, and it finds many applications in various scientific computing areas. A major advantage of the ALM is that the step for updating the dual variable can be further relaxed with a step size in ( 0 , 2 ) (0,2) , and this advantage can easily lead to numerical acceleration for the ALM. When a separable convex programming problem is discussed and a corresponding splitting version of the classic ALM is considered, convergence may not be guaranteed and thus it is seemingly impossible that a step size in ( 0 , 2 ) (0,2) can be carried on to the relaxation step for updating the dual variable. We show that for a parallel splitting version of the ALM, a step size in ( 0 , 2 ) (0,2) can be maintained for further relaxing both the primal and dual variables if the relaxation step is simply corrected by a rank-two matrix. Hence, a rank-two relaxed parallel splitting version of the ALM with a step size in ( 0 , 2 ) (0,2) is proposed for separable convex programming problems. We validate that the new algorithm can numerically outperform existing algorithms of the same kind significantly by testing some applications.

A dual-primal balanced augmented Lagrangian method for linearly constrained convex programming

Most recently, a balanced augmented Lagrangian method (ALM) has been proposed by He and Yuan for the canonical convex minimization problem with linear constraints, which advances the original ALM by balancing its subproblems, improving its implementation and enlarging its applicable range. In this paper, we propose a dual-primal version of the newly developed balanced ALM, which updates the new iterate via a conversely dual-primal iterative order formally. The new algorithm inherits all advantages of the prototype balanced ALM, and it can be extended to more general separable convex programming problems with both linear equality and inequality constraints. The convergence analysis of the proposed method can be well conducted in the context of variational inequalities. In particular, by some application problems, we numerically validate that these balanced ALM type methods can outperform existing algorithms of the same kind significantly.

A faster generalized ADMM-based algorithm using a sequential updating scheme with relaxed step sizes for multiple-block linearly constrained separable convex programming

The multi-block linearly constrained separable convex optimization is frequently applied in numerous applications, including image/signal processing, statistical learning, and data mining, where the objective function is the sum of multiple individual convex functions, and the key constraints are linear. A classical approach to solving such optimization problem could be the alternating direction method of multipliers (ADMM). It decomposes the subproblem into a series of small-scale ones such that its per-iteration cost may be meager. ADMM, however, is designed initially for the two-block model, and its convergence cannot be guaranteed for a general multi-block model without additional assumptions. Dai et al. (2017) proposed the algorithm SUSLM (for Sequential Updating Scheme of the Lagrange Multiplier) for separable convex programming problems. The Lagrange multipliers are updated several times at each iteration, and a correction step is imposed at the end of each iteration. In order to derive its convergence property, a correction step is imposed at the end of each iteration. In this paper, we improve the SUSLM algorithm by introducing two controlled parameters in the updating expressions for decision variables and Lagrange multipliers. The condition of step sizes is then relaxed. We show experimentally that our SUSLM algorithm converges faster than SUSLM. Moreover, result comparisons on robust principal component analysis (RPCA) show better performances than other ADMM-based algorithms.

Convergence study of indefinite proximal ADMM with a relaxation factor

The alternating direction method of multipliers (ADMM) is widely used to solve separable convex programming problems. At each iteration, the classical ADMM solves the subproblems exactly. For many problems arising from practical applications, it is usually impossible or too expensive to obtain the exact solution of a subproblem. To overcome this, a special proximal term is added to ease the solvability of a subproblem. In the literature, the proximal term can be relaxed to be indefinite while still with a convergence guarantee; this relaxation permits the adoption of larger step sizes to solve the subproblem, which particularly accelerates its performance. A large value of the relaxation factor introduced in the dual step of ADMM also plays a vital role in accelerating its performance. However, it is still not clear whether these two acceleration strategies can be used simultaneously with no restriction on the penalty parameter. In this paper, we answer this question affirmatively and conduct a rigorous convergence analysis for indefinite proximal ADMM with a relaxation factor (IP-ADMM $$_{r}$$ ), reveal the relationships between the parameter in the indefinite proximal term and the relaxation factor to ensure its global convergence, and establish the worst-case convergence rate in the ergodic sense. Finally, some numerical results on basis pursuit and total variation-based denoising with box constraint problems are presented to verify the efficiency of IP-ADMM $$_{r}$$ .

A Proximal Alternating Direction Method of Multipliers with a Substitution Procedure

In this paper, we considers the separable convex programming problem with linear constraints. Its objective function is the sum of m individual blocks with nonoverlapping variables and each block consists of two functions: one is smooth convex and the other one is convex. For the general case m≥3, we present a gradient-based alternating direction method of multipliers with a substitution. For the proposed algorithm, we prove its convergence via the analytic framework of contractive-type methods and derive a worst-case O1/t convergence rate in nonergodic sense. Finally, some preliminary numerical results are reported to support the efficiency of the proposed algorithm.

Efficient Secure Outsourcing of Large-Scale Convex Separable Programming for Big Data

Big data has become a key basis of innovation and intelligence, potentially making our lives more convenient and bringing new opportunities to the modern society. Towards this goal, a critical underlying task is to solve a series of large-scale fundamental problems. Conducting such large-scale data analytics in a timely manner requires a large amount of computing resources, which may not be available for individuals and small companies in practice. By outsourcing their computations to the cloud, clients can solve such problems in a cost-effective way. However, confidential data stored at the cloud is vulnerable to cyber attacks, and thus needs to be protected. Previous works employ cryptographic techniques like homomorphic encryption, which significantly increase the computational complexity of solving a large-scale problem at the cloud and is impractical for big data applications. For the first time in the literature, we present an efficient secure outsourcing scheme for convex separable programming problems (CSPs). In particular, we first develop efficient matrix and vector transformation schemes only based on arithmetic operations that are computationally indistinguishable both in value and in structure under a chosen-plaintext attack (CPA). Then, we design a secure outsourcing scheme in which the client and the cloud collaboratively solve the transformed problems. The client can efficiently verify the correctness of returned results to prevent any malicious behavior of the cloud. Theoretical correctness and privacy analysis together show that the proposed scheme obtains optimal results and that the cloud cannot learn private information from the client's concealed data. We conduct extensive simulations on Amazon Elastic Cloud Computing (EC2) platform and find that our proposed scheme provides significant time savings to the clients.

Separable programming problems with the max-product fuzzy relation equation constraints

In this paper, the separable programming problem subject to Fuzzy Relation Equation (FRE) constraints is studied. It is decomposed to two subproblems with decreasing and increasing objective functions with the same constraints. They are solved by the maximum solution and one of minimal solutions of its feasible domain, respectively. Their combination produces the original optimal solution. The detection of the optimal solution of the second subproblem by finding all the minimal solutions will be very time-consuming because of its NP-hardness. To overcome such difficulty, two types of sufficient conditions are proposed to find some of its optimal components or all of them. Under the first type sufficient conditions, some procedures are given to simplify the original problem. Also, a value matrix is defined and an algorithm is proposed to compute an initial upper bound on its optimal objective value using the matrix. Then, a branch-and-bound method is extended using the matrix and initial upper bound to solve the simplified problem without finding all the minimal solutions.

Explicit model of dual programming and solving method for a class of separable convex programming problems

ABSTRACTAn objective function for a dual model of nonlinear programming problems is an implicit function with respect to Lagrangian multipliers. This study aims to address separable convex programming problems. An explicit expression with respect to Lagrangian multipliers is derived for the dual objective function. The exact solution of the dual model can be achieved because an explicit objective function is more exact than an approximated objective function. Then, a set of improved Lagrangian multipliers can be used to obtain the optimal solution of the original nonlinear programming model. A corresponding dual programming and explicit model (DP-EM) method is proposed and applied to the structural topology optimization of continuum structures. The solution efficiency of the DPEM is compared with the dual sequential quadratic programming (DSQP) method and method of moving asymptotes (MMA). The results show that the DP-EM method is more efficient than the DSQP and MMA.

Convergence of ADMM for Three-Block Separable Quadratic Programming Problems with Linear Constraints

Convergence of ADMM for Three-Block Separable Quadratic Programming Problems with Linear Constraints

A new descent alternating direction method with LQP regularization for the structured variational inequalities

In this paper, we suggest and analyze a new logarithmic–quadratic proximal alternating direction scheme for the separable constrained convex programming problem. The main contribution of this paper, the predictor is obtained via solving LQP system approximately under significantly relaxed accuracy criterion and the new iterate is obtained by using a new direction with a new step size $$\alpha _k$$ . Global convergence of the proposed method is proved under certain assumptions. We also reported some numerical results to illustrate the efficiency of the proposed method.

SQP alternating direction method with a new optimal step size for solving variational inequality problems with separable structure

In this paper, we suggest and analyze a new alternating direction scheme for the separable constrained convex programming problem. The theme of this paper is twofold. First, we consider the square-quadratic proximal (SQP) method. Next, by combining the alternating direction method with SQP method, we propose a descent SQP alternating direction method by using the same descent direction as in [6] with a new step size ?k. Under appropriate conditions, the global convergence of the proposed method is proved. We show the O(1/t) convergence rate for the SQP alternating direction method. Some preliminary computational results are given to illustrate the efficiency of the proposed method.

A parameterized proximal point algorithm for separable convex optimization

In this paper, we develop a parameterized proximal point algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent with a worst-case O(1/t) convergence rate, wheret denotes the iteration number. By properly choosing the algorithm parameters, numerical experiments on solving a sparse optimization problem arising from statistical learning show that our P-PPA could perform significantly better than other state-of-the-art methods, such as the alternating direction method of multipliers and the relaxed proximal point algorithm.

A method to solve separable fuzzy nonlinear programming problem

This paper formulates piecewise fuzzy linear approximation of fuzzy nonlinear function of a single variable. First, we construct the equation of fuzzy line segment through two fuzzy points using -cuts of two fuzzy points. Now the fuzzy nonlinear function is represented by a set of connected fuzzy line segments. Fuzzy line segments have been defined by partitioning the domain of a fuzzy nonlinear function. Then formulation of approximating fuzzy linear programming problem has been given to find the solution of fuzzy nonlinear separable programming problem. An illustrative example is given to demonstrate the methodology.

A method to solve separable fuzzy nonlinear programming problem

A method to solve separable fuzzy nonlinear programming problem

A new LQP alternating direction method for solving variational inequality problems with separable structure

We presented a new logarithmic-quadratic proximal alternating direction scheme for the separable constrained convex programming problem. The predictor is obtained by solving series of related systems of non-linear equations in a parallel wise. The new iterate is obtained by searching the optimal step size along a new descent direction. The new direction is obtained by the linear combination of two descent directions. Global convergence of the proposed method is proved under certain assumptions. We show the O(1 / t) convergence rate for the parallel LQP alternating direction method.

A parametric simplex algorithm for biobjective piecewise linear programming problems

In this paper we attempt to develop a parametric simplex algorithm for solving biobjective convex separable piecewise linear programming problems. The algorithm presented in this paper can be regarded as an extension of the parametric simplex algorithm for solving biobjective linear programming problems to the piecewise linear case. Analogous to the linear case, this parametric simplex algorithm provides a decomposition of parametric space. A numerical example is presented to illustrate the implementation of the 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.