Nonsmooth Unconstrained Optimization Problems Research Articles

A new spectral conjugate subgradient method with application in computed tomography image reconstruction

A new spectral conjugate subgradient method is presented to solve nonsmooth unconstrained optimization problems. The method combines the spectral conjugate gradient method for smooth problems with the spectral subgradient method for nonsmooth problems. We study the effect of two different choices of line search, as well as three formulas for determining the conjugate directions. In addition to numerical experiments with standard nonsmooth test problems, we also apply the method to several image reconstruction problems in computed tomography, using total variation regularization. Performance profiles are used to compare the performance of the algorithm using different line search strategies and conjugate directions to that of the original spectral subgradient method. Our results show that the spectral conjugate subgradient algorithm outperforms the original spectral subgradient method, and that the use of the Polak–Ribière formula for conjugate directions provides the best and most robust performance.

Stochastic radiation field optimization for microwave staring correlated imaging via spatial correlation minimization

Microwave staring correlated imaging (MSCI) is a staring high-resolution microwave imaging technique, employing the temporal–spatial stochastic radiation field (TSSRF). The imaging capability of MSCI depends on the spatial correlation of the TSSRF, which is equivalent to the incoherence of the sensing matrix in discrete form. A waveform design method for MSCI using multifrequency signals to reduce the spatial correlation of the TSSRF is proposed. The amplitude of each frequency component in the multifrequency signals serves as the design variable. Considering the power and the amplitude constraints in the practical radar system, we first form the waveform design as a constrained optimization problem. Then, it is reformulated as a nonsmooth unconstrained optimization problem by the penalty function method. With the smoothing approximation, the reformulated problem is solved by alternating minimization. Numerical experiments show that the proposed method can decrease the spatial correlation of the TSSRF and improve the imaging performance of MSCI as a result.

A numerical study of applying spectral-step subgradient method for solving nonsmooth unconstrained optimization problems

The purpose of this work is two-fold. First, we present the spectral-step subgradient method to solve nonsmooth unconstrained optimization problems. It combines the classical subgradient approach and a nonmonotone linesearch with the spectral step length, which does not require any previous knowledge of the optimal value. We focus on the interesting case in which the objective function is convex and continuously differentiable almost everywhere, and it is often non-differentiable at minimizers. Secondly, we use performance profiles to compare the spectral-step subgradient method with other subgradient methods. This comparison will allow us to place the spectral-step subgradient algorithm among other subgradient algorithms.

An effective adaptive trust region algorithm for nonsmooth minimization

In this paper, an adaptive trust region algorithm that uses Moreau–Yosida regularization is proposed for solving nonsmooth unconstrained optimization problems. The proposed algorithm combines a modified secant equation with the BFGS update formula and an adaptive trust region radius, and the new trust region radius utilizes not only the function information but also the gradient information. The global convergence and the local superlinear convergence of the proposed algorithm are proven under suitable conditions. Finally, the preliminary results from comparing the proposed algorithm with some existing algorithms using numerical experiments reveal that the proposed algorithm is quite promising for solving nonsmooth unconstrained optimization problems.

Numerical infinitesimals in a variable metric method for convex nonsmooth optimization

The objective of the paper is to evaluate the impact of the infinity computing paradigm on practical solution of nonsmooth unconstrained optimization problems, where the objective function is assumed to be convex and not necessarily differentiable. For such family of problems, the occurrence of discontinuities in the derivatives may result in failures of the algorithms suited for smooth problems.We focus on a family of nonsmooth optimization methods based on a variable metric approach, and we use the infinity computing techniques for numerically dealing with some quantities which can assume values arbitrarily small or large, as a consequence of nonsmoothness. In particular we consider the case, treated in the literature, where the metric is defined via a diagonal matrix with positive entries.We provide the computational results of our implementation on a set of benchmark test-problems from scientific literature.

Complementarity Techniques for Minimal Coordinate Contact Dynamics

In this paper, nonsmooth contact dynamics of articulated rigid multibody systems is formulated as a complementarity problem. Minimal coordinate (MC) formulation is used to derive the dynamic equations of motion as it provides significant computational cost benefits and leads to a smaller-sized complementarity problem when compared with the frequently used redundant coordinate (RC) formulation. Additionally, an operational space (OS) formulation is employed to take advantage of the low-order structure-based recursive algorithms that do not require mass matrix inversion, leading to a further reduction in these computational costs. Based on the accuracy with which Coulomb's friction cone is modeled, the complementarity problem can be posed either as a linear complementarity problem (LCP), where the friction cone is approximated using a polygon, or as a nonlinear complementarity problem (NCP), where the friction cone is modeled exactly. Both formulations are studied in this paper. These complementarity problems are further recast as nonsmooth unconstrained optimization problems, which are solved by employing a class of Levenberg–Marquardt (LM) algorithms. The necessary theory detailing these techniques is discussed and five solvers are implemented to solve contact dynamics problems. A simple test case of a sphere moving on a plane surface is used to validate these solvers for a single contact, whereas a 12-link complex pendulum example is chosen to compare the accuracy of the solvers for the case of multiple simultaneous contacts. The simulation results validate the MC-based NCP formulations developed in this paper. Moreover, we observe that the LCP solvers deliver accuracy comparable to that of the NCP solvers when the friction cone direction vectors in the contact tangent plane are aligned with the sliding contact velocity at each time step. The theory and simulation results show that the NCP approach can be seamlessly recast into an MC OS formulation, thus allowing for accurate modeling of frictional contacts, while at the same time reducing overall computational costs associated with contact and collision dynamics problems in articulated rigid body systems.

A modified tunneling function method for non-smooth global optimization and its application in artificial neural network

For solving a class of non-smooth unconstrained global optimization problems, we present a novel definition of the modified tunneling function which combines the characters of tunneling function and filled function, and then give a one-parameter modified tunneling function. Issues covered in the presented work include: theoretical properties, solution algorithms and numerical experiments. Furthermore, an improved artificial neural network hydrological forecasting method using the modified tunneling function is also reported. The preliminary experiment results confirm that the proposed approach is promising.

A quasi-Newton method for unconstrained non-smooth problems

We present a method, based on a variational problem, for solving a non-smooth unconstrained optimization problem. We assume that the objective function is a Lipschitz continuous and a regular function. In this case the function of our variational problem is semismooth and a quasi-Newton method may be used to solve the variational problem. A convergence theorem for our algorithm and its discrete version is also proved. Preliminary computational results show that the method performs quite well and can compete with other methods.

Issuing for perishable inventory management with a minimum inventory volume constraint

We study a perishable inventory system that requires to maintain a minimum inventory volume at all times, where the minimum amount in the constraint is significant with respect to regular market demand and the traditional Economic Manufacturing Quantity (EMQ) models do not suffice. The problem is motivated by applications in homeland security, which are related to the management of pharmaceuticals in the Strategic National Stockpile (SNS) for emergency preparedness. We use a modified EMQ model to represent the system and consider the issuing policy given a fixed ordering quantity, as well as the joint ordering and issuing policy problem. We aim to maximize the profit of the system under a linearly-decreasing price structure assumption. We first present the optimal issuing policy with a given ordering quantity for this modified EMQ model. Then we demonstrate that maximizing the revenue of the ordering policy with FIFO (first-in, first-out) and LIFO (last-in, first-out) issuing policies can be formulated as a non-convex non-smooth unconstrained optimization problem. The properties of both problems (e.g. optimizing issuing policy with a fixed ordering quantity and optimizing ordering policy with a fixed issuing policy) is analyzed and an efficient exact algorithm is presented to solve the joint ordering and issuing problem. We show that the LIFO issuing policy is optimal with a linearly-decreasing price structure that deteriorates with the age of the item.

A modified Polak–Ribière–Polyak conjugate gradient algorithm for nonsmooth convex programs

The conjugate gradient (CG) method is one of the most popular methods for solving smooth unconstrained optimization problems due to its simplicity and low memory requirement. However, the usage of CG methods is mainly restricted to solving smooth optimization problems so far. The purpose of this paper is to present efficient conjugate gradient-type methods to solve nonsmooth optimization problems. By using the Moreau–Yosida regulation (smoothing) approach and a nonmonotone line search technique, we propose a modified Polak–Ribière–Polyak (PRP) CG algorithm for solving a nonsmooth unconstrained convex minimization problem. Our algorithm possesses the following three desired properties. (i) The search direction satisfies the sufficient descent property and belongs to a trust region automatically; (ii) the search direction makes use of not only the gradient information but also the function value information; and (iii) the algorithm inherits an important property of the well-known PRP method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, preventing a sequence of tiny steps from happening. Under standard conditions, we show that the algorithm converges globally to an optimal solution. Numerical experiment shows that our algorithm is effective and suitable for solving large-scale nonsmooth unconstrained convex optimization problems.

A log-barrier method for mesh quality improvement and untangling

The presence of a few inverted or poor-quality mesh elements can negatively affect the stability, convergence and efficiency of a finite element solver and the accuracy of the associated partial differential equation solution. We propose a mesh quality improvement and untangling method that untangles a mesh with inverted elements and improves its quality. Worst element mesh quality improvement and untangling can be formulated as a nonsmooth unconstrained optimization problem, which can be reformulated as a smooth constrained optimization problem. Our technique solves the latter problem using a log-barrier interior point method and uses the gradient of the objective function to efficiently converge to a stationary point. The method uses a logarithmic barrier function and performs global mesh quality improvement. We have also developed a smooth quality metric that takes both signed area and the shape of an element into account. This quality metric assigns a negative value to an inverted element. It is used with our algorithm to untangle a mesh by improving the quality of an inverted element to a positive value. Our method usually yields better quality meshes than existing methods for improvement of the worst quality elements, such as the active set, pattern search, and multidirectional search mesh quality improvement methods. Our method is faster and more robust than existing methods for mesh untangling, such as the iterative stiffening method.

Global minimization of non-smooth unconstrained problems with filled function

For smooth or non-smooth unconstrained global optimization problems, an one parameter filled function is derived to identify their global optimizers or approximately global optimizers. The theoretical properties of the proposed function are investigated. Based on the filled function, an algorithm is designed for solving unconstrained global optimization problems. The algorithm consists of two phases: local minimization and filling. The former is intended to minimize the objective function and obtain a local optimizer, the latter aims to find a better initial point for the first phase. Numerical experimentation is also provided. The preliminary computational results confirm that the proposed filled function approach is promising.

Perishable inventory management system with a minimum volume constraint

The federal government maintains large quantities of medical supplies in stock as part of its Strategic National Stockpile (SNS) to protect the American public in case of a public health emergency. Managing these large perishable inventories effectively can help reduce the cost of the SNS and improves national security. In this paper, we propose a modified Economic Manufacturing Quantity (EMQ) model for perishable inventory with a minimum volume constraint, which is applicable to managing the inventory of medicines for the Strategic National Stockpile. We demonstrate that minimizing the cost of maintaining such a system can be formulated as a non-convex non-smooth unconstrained optimization problem. The property of this model is analysed and an efficient exact algorithm is presented to solve this problem. In the numerical experiment part, we perform sensitivity analysis on several government-controlled system parameters to illustrate how the government can obtain lower costs or a larger stockpile at the same cost by allowing more freedom in the management of the stocks.

A method for non-differentiable optimization problems

We introduce a new method for solving a class of non-smooth unconstrained optimization problems. The method constructs a sequence<texlscub>x k </texlscub>in ℝ n , and at the kth iteration, a search direction h k is considered, where h k is a solution to a variational inequality problem. A convergence theorem for our algorithm model and its discrete version can be easily proved. Furthermore, preliminary computational results show that the new method performs quite well and can compete with other methods.

CONTINUOUS OPTIMIZATION APPROACHES TO THE IDENTIFICATION OF PIECEWISE AFFINE SYSTEMS

Approaches for the identification of piecewise affine systems are presented. The identification problem for piecewise affine systems is formulated as a smooth constrained, and a nonsmooth unconstrained optimization problem. The main advantage of the new problem formulations is the fact that no mixed integer problems have to be solved within this method. The feasibility and performance of the procedures are demonstrated by identifying two piecewise affine characteristic maps.

Global Interval Methods for Local Nonsmooth Optimization

An interval method for determining local solutions of nonsmooth unconstrained optimization problems is discussed. The objective function is assumed to be locally Lipschitz and to have appropriate interval inclusions. The method consists of two parts, a local search and a global continuation and termination. The local search consists of a globally convergent descent algorithm showing similarities to e-bundle methods. While e-bundle methods use polytopes as inner approximations of the e-subdifferentials, which are the main tools of almost all bundle concepts, our method uses axes parallel boxes as outer approximations of the e-subdifferentials. The boxes are determined almost automatically with inclusion techniques of interval arithmetic. The dimension of the boxes is equal to the dimension of the problem and remains constant during the whole computation. The application of boxes does not suffer from the necessity to invest methodical and computational efforts to adapt the polytopes to the latest state of the computation as well as to simplify them when the number of vertices becomes too large, as is the case with the polytopes. The second part of the method applies interval techniques of global optimization to the approximative local solution obtained from the search of the first part in order to determine guaranteed error bounds or to improve the solution if necessary. We present prototype algorithms for both parts of the method as well as a complete convergence theory for them and demonstrate how outer approximations can be obtained.

A trust region algorithm for bilevel programing problems

A trust region algorithm is proposed for solving bilevel programming problems where the lower lwel programming problem is a strongly convex programming problem with linear constraints. This algorithm is based on a trust region algorithm for nonsmooth unconstrained optimization problems, and its global convergence is also proved.

Monotone generalized differentiability in nonsmooth optimization

The paper deals with mathematical programming and algebraic problems, by formulating both ones as nonsmooth unconstrained optimization problems. Therefore minimization algorithms, using monotone generalized subdifferentials, are define and numerical results reported, to show their performance.