Convex Nonlinear Programming Research Articles

Moreau–Yosida regularization of Lagrangian-dual functions for a class of convex optimization problems

In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems, which originates from multi-stage stochastic convex nonlinear programs. We study the Moreau---Yosida regularization of the Lagrangian-dual function and prove that the regularized function ? is piecewise C 2, in addition to the known smoothness property. This property is then used to investigate the semismoothness of the gradient mapping of the regularized function. Finally, we show that the Clarke generalized Jacobian of the gradient mapping is BD-regular under some conditions.

Low-Rank Optimization on the Cone of Positive Semidefinite Matrices

We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of $X$. It is thus very effective for solving programs that have a low rank solution. The factorization $X=Y Y^T$ evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem.

Arrival Rate Approximation by Nonnegative Cubic Splines

We describe an optimization method to approximate the arrival-rate function of a nonhomogeneous Poisson process based on observed arrival data. We estimate the function by cubic splines, using an optimization model based on the maximum-likelihood principle. A critical feature of the model is that the splines are constrained to be nonnegative everywhere. We enforce these constraints by using a characterization of nonnegative polynomials by positive semidefinite matrices. We also describe versions of our model that allow for periodic arrival-rate functions and input data of limited time precision. We formulate the estimation problem as a convex nonlinear program, and solve it with standard nonlinear optimization packages. We present numerical results using both an actual record of e-mail arrivals over a period of 60 weeks, and artificially generated data sets. We also present a cross-validation procedure for determining an appropriate number of spline knots to model a set of arrival observations.

Optimal batch sizing in a multi-stage production system with rework consideration

In a production system, rework process plays an important role in eliminating waste and effectively controlling the cost of manufacturing. Determining the optimal batch size in a system that allows for rework is, therefore, a worthwhile objective to minimize the inventory cost of work-in-processes and the finished goods. In this paper, models for the optimum batch quantity in a multi-stage system with rework process have been developed for two different operational policies. Policy 1 deals with the rework within the same cycle with no shortage and policy 2 deals with the rework done after N cycles, incurring shortages in each cycle. The major components that play a role in minimizing this cost of the system are manufacturing setups, work-in-processes, storage of finished goods, rework processing, waiting-time, and penalty costs to discourage the generation of defectives. The mathematical structure of this rework processing model falls under a nonlinear convex programming problems for which a closed-form solution has been proposed and results are demonstrated through numerical examples, followed by sensitivity analyses of different important parameters. It is concluded that the total cost in policy 2 tends to be smaller than that in policy 1 at lower proportion of defectives if the in-process carrying cost is low. Policy 2 may be preferred when the work-in-process carrying cost is low and the penalty cost is negligible.

A polynomial path following algorithm for convex programming

In this paper, based on combined homotopy interior point method we propose an interior point algorithm for convex nonlinear programming. The algorithm ensures that the obtained iterative points are interior points of the feasible set in terms of the technique of β-cone neighborhood. We establish the global convergence of the algorithm. Furthermore, it is shown that the algorithm has O ( n L ) iteration complexity. The preliminary numerical experiments indicate that the algorithm is efficient.

A discrete-time neural network for solving nonlinear convex problems with hybrid constraints

This paper investigates a discrete-time neural network model for solving nonlinear convex programming problems with hybrid constraints. The neural network finds the solution of both primal and dual problems and converges to the corresponding exact solution globally. We prove here that the proposed neural network is globally exponentially stable. Furthermore, we extend the proposed neural network for solving a class of monotone variational inequality problems with hybrid constraints. Compared with other existing neural networks for solving such problems, the proposed neural network has a low complexity for implementation without a penalty parameter and converge an exact solution to convex problem with hybrid constraints. Some numerical simulations for justifying the theoretical analysis are also given. The numerical simulations are shown that in the new model note only the cost of the hardware implementation is not relatively expensive, but also accuracy of the solution is greatly good.

Geometric dual formulation for first-derivative-based univariate cubic L 1 splines

With the objective of generating "shape-preserving" smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based $$\mathcal{C}^1$$ -smooth univariate cubic L 1 splines. An L 1 spline minimizes the L 1 norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L 1 spline is a nonsmooth non-linear convex program. Via Fenchel's conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.

An self-adaptive LQP method for constrained variational inequalities

In this paper, we present a logarithmic-quadratic proximal (LQP) type prediction–correction methods for solving constrained variational inequalities VI ( S , f ) , where S is a convex set with linear constraints. The computational load in each iteration is quite tiny. However, the number of iterations is significantly dependent on a parameter which balances the primal and dual variables. We then propose a self-adaptive prediction–correction method that adjusts the scalar parameter automatically. Under certain conditions, the global convergence of the proposed method is established. In order to demonstrate the efficiency of the proposed method, we provide numerical results for a convex nonlinear programming and traffic equilibrium problems.

A sequential cutting plane algorithm for solving convex NLP problems

In this paper we look at a new algorithm for solving convex nonlinear programming optimization problems. The algorithm is a cutting plane-based method, where the sizes of the subproblems remain fixed, thus avoiding the issue with constantly growing subproblems we have for the classical Kelley’s cutting plane algorithm. Initial numerical experiments indicate that the algorithm is considerably faster than Kelley’s cutting plane algorithm and also competitive with existing nonlinear programming algorithms.

Subgradient-based feedback neural networks for non-differentiable convex optimization problems

This paper developed the dynamic feedback neural network model to solve the convex nonlinear programming problem proposed by Leung et al. and introduced subgradient-based dynamic feedback neural networks to solve non-differentiable convex optimization problems. For unconstrained non-differentiable convex optimization problem, on the assumption that the objective function is convex coercive, we proved that with arbitrarily given initial value, the trajectory of the feedback neural network constructed by a projection subgradient converges to an asymptotically stable equilibrium point which is also an optimal solution of the primal unconstrained problem. For constrained non-differentiable convex optimization problem, on the assumption that the objective function is convex coercive and the constraint functions are convex also, the energy functions sequence and corresponding dynamic feedback subneural network models based on a projection subgradient are successively constructed respectively, the convergence theorem is then obtained and the stopping condition is given. Furthermore, the effective algorithms are designed and some simulation experiments are illustrated.

An Algorithm for Approximate Multiparametric Convex Programming

For multiparametric convex nonlinear programming problems we propose a recursive algorithm for approximating, within a given suboptimality tolerance, the value function and an optimizer as functions of the parameters. The approximate solution is expressed as a piecewise affine function over a simplicial partition of a subset of the feasible parameters, and it is organized over a tree structure for efficiency of evaluation. Adaptations of the algorithm to deal with multiparametric semidefinite programming and multiparametric geometric programming are provided and exemplified. The approach is relevant for real-time implementation of several optimization-based feedback control strategies.

Performance enhancement of adaptive orthogonal modulation in wireless CDMA systems

Recent research in wireless code-division multiple-access systems has shown that adaptive rate/power control can considerably increase network throughput relative to systems that use only power or rate control. In this paper, we consider joint power/rate optimization in the context of orthogonal modulation (OM) and investigate the additional performance gains achieved through adaptation of the OM order. We show that such adaptation can significantly increase network throughput, while simultaneously reducing the per-bit energy consumption relative to fixed-order modulation systems. The optimization is carried out under two different objective functions: minimizing the maximum service time and maximizing the sum of user rates. For the first objective function, we prove that the optimization problem can be formulated as a generalized geometric program (GGP). We then show how this GGP can be transformed into a nonlinear convex program, which can be solved optimally and efficiently. For the second objective function, we obtain a lower bound on the performance gain of adaptive OM (AOM) over fixed-modulation systems. Numerical results indicate that relative to an optimal joint rate/power control fixed-order modulation scheme, the proposed AOM scheme achieves significant throughput and energy gains.

Certain questions in solving block nonlinear optimization problems with coupling variables

A decomposition scheme for block nonlinear convex programming problems with coupling variables is considered. The possibility is examined of using approximated solutions of subproblems for generating subgradients of the functions that appear in the master problem. A regularization of the original problem that simplifies the master problem is considered.

On the complexity of a combined homotopy interior method for convex programming

In [G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush–Kuhn–Tucker point of a nonconvex programming problem, Nonlinear Anal. 32 (1998) 761–768; G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics, Proceedings of the Second Japan–China Seminar on Numerical Mathematics, Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9–16; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Appl. Math. Comput. 84 (1997) 193–211.], a combined homotopy was constructed for solving non-convex programming and convex programming with weaker conditions, without assuming the logarithmic barrier function to be strictly convex and the solution set to be bounded. It was proven that a smooth interior path from an interior point of the feasible set to a K–K–T point of the problem exists. This shows that combined homotopy interior point methods can solve the problem that commonly used interior point methods cannot solve. However, so far, there is no result on its complexity, even for linear programming. The main difficulty is that the objective function is not monotonically decreasing on the combined homotopy path. In this paper, by taking a piecewise technique, under commonly used conditions, polynomiality of a combined homotopy interior point method is given for convex nonlinear programming.

A Disjunctive Cutting-Plane-Based Branch-and-Cut Algorithm for 0−1 Mixed-Integer Convex Nonlinear Programs

In this paper, a disjunctive cutting-plane-based branch-and-cut algorithm is developed to solve the 0−1 mixed-integer convex nonlinear programming (MINLP) problems. In a branch-and-bound framework, the 0−1 MINLP problem is approximated with a 0−1 mixed-integer linear program at each node, and then the lift-and-project technology is used to generate valid cuts to accelerate the branching process. The cut is produced by solving a linear program that is transformed from a projection problem, in terms of the disjunction on a free binary variable, and its dual solutions are applied to lift the cut to become valid throughout the enumeration tree. A strengthening process is derived to improve the coefficients of the cut by imposing integrality on the left free binary variables. Finally, the computational results on four test problems indicate that the added cutting planes can reduce the branching process greatly and show that the proposed algorithm is very promising for large-scale 0−1 MINLP problems, because a linear program is always computationally less expensive than a nonlinear program.

Lagrangian Coordination and Analytical Target Cascading: Solving ATC-Decomposed Problems with Lagrangian Duality

Lagrangian duality is a powerful tool for dealing with large mathematical programs which require decomposition. We consider a large-scale convex nonlinear program that is decomposed according to the scheme of analytical target cascading and propose a Lagrangian duality-based coordination in which solutions of resulting subproblems converge to a solution of the original problem. We present a subgradient algorithm to achieve said solution, demonstrate with an example, and conclude with an extension to multiple level problems with multiple subsystems on each level.

A Recurrent Neural Network for Solving Nonlinear Convex Programs Subject to Linear Constraints

In this paper, we propose a recurrent neural network for solving nonlinear convex programming problems with linear constraints. The proposed neural network has a simpler structure and a lower complexity for implementation than the existing neural networks for solving such problems. It is shown here that the proposed neural network is stable in the sense of Lyapunov and globally convergent to an optimal solution within a finite time under the condition that the objective function is strictly convex. Compared with the existing convergence results, the present results do not require Lipschitz continuity condition on the objective function. Finally, examples are provided to show the applicability of the proposed neural network.

Necessary and Sufficient Conditions for Weak No-Arbitrage in Securities Markets with Frictions

In this paper we consider a financial market model with frictions which include transaction costs, bid-ask spread and taxes. By using optimization, linear and nonlinear programming and convex programming techniques, several necessary and sufficient conditions are derived for the weak no-arbitrage. Some results on state prices are also provided. The results of this paper can provide at least some theoretical insight to the problem.

A new nonlinear neural network for solving convex nonlinear programming problems

This paper presents a new recurrent neural network for solving convex nonlinear programming problems. The new model is simpler and more intuitive than existing models and converge very fast to exact solution of the original problem. We show that this new model is asymptotically stable.

A modified augmented Lagrangian method for a class of monotone variational inequalities

The augmented Lagrangian method is an attractive approach for solving linearly constrained convex optimization problems (CCOP). The method is iterative and the task in each iteration is to solve a (convex optimization) sub-problem whose dimension is equal to the number of variables. For large problems, solving such a sub-problem can be still very expensive. In this paper, we propose a modified augmented Lagrangian method for a class of monotone variational inequalities which include CCOP. The computational load in each iteration of the proposed method is quite small. In order to demonstrate the ability and efficiency of the proposed method, we present some numerical results for convex nonlinear programming and traffic network equilibrium problems.