Convex Nonlinear Programming Research Articles

Extensions of Dinkelbach's algorithm for solving non-linear fractional programming problems

Dinkelbach's algorithm was developed to solve convex fractinal programming. This method achieves the optimal solution of the optimisation problem by means of solving a sequence of non-linear convex programming subproblems defined by a parameter. In this paper it is shown that Dinkelbach's algorithm can be used to solve general fractional programming. The applicability of the algorithm will depend on the possibility of solving the subproblems. Dinkelbach's extended algorithm is a framework to describe several algorithms which have been proposed to solve linear fractional programming, integer linear fractional programming, convex fractional programming and to generate new algorithms. The applicability of new cases as nondifferentiable fractional programming and quadratic fractional programming has been studied. We have proposed two modifications to improve the speed-up of Dinkelbachs algorithm. One is to use interpolation formulae to update the parameter which defined the subproblem and another truncates the solution of the suproblem. We give sufficient conditions for the convergence of these modifications. Computational experiments in linear fractional programming, integer linear fractional programming and non-linear fractional programming to evaluate the efficiency of these methods have been carried out.

Control applications of nonlinear convex programming

Since 1984 there has been a concentrated effort to develop efficient interior-point methods for linear programming (LP). In the last few years researchers have begun to appreciate a very important property of these interior-point methods (beyond their efficiency for LP): they extend gracefully to nonlinear convex optimization problems. New interior-point algorithms for problem classes such as semidefinite programming (SDP) or second-order cone programming (SOCP) are now approaching the extreme efficiency of modern linear programming codes. In this paper we discuss three examples of areas of control where our ability to efficiently solve nonlinear convex optimization problems opens up new applications. In the first example we show how SOCP can be used to solve robust open-loop optimal control problems. In the second example, we show how SOCP can be used to simultaneously design the set-point and feedback gains for a controller, and compare this method with the more standard approach. Our final application concerns analysis and synthesis via linear matrix inequalities and SDP.

Analog neural nonderivative optimizers

Continuous-time neural networks for solving convex nonlinear unconstrained programming problems without using gradient information of the objective function are proposed and analyzed. Thus, the proposed networks are nonderivative optimizers. First, networks for optimizing objective functions of one variable are discussed. Then, an existing one-dimensional optimizer is analyzed, and a new line search optimizer is proposed. It is shown that the proposed optimizer network is robust in the sense that it has disturbance rejection property. The network can be implemented easily in hardware using standard circuit elements. The one-dimensional net is used as a building block in multidimensional networks for optimizing objective functions of several variables. The multidimensional nets implement a continuous version of the coordinate descent method.

A combined homotopy interior point method for convex nonlinear programming

In this paper, we present a new interior point method—combined homotopy interior point method (CHIP method) for convex nonlinear programming. Without strict convexity of the logarithmic barrier function and boundedness and nonemptiness of the solution set, we prove that for any ϵ > 0, an ϵ-solution of the problem can be obtained by the CHIP method. To our knowledge, strict convexity of the logarithmic barrier function and nonemptiness and boundedness of the solution set are the essential assumptions of the well-known center path-following method. Therefore, the CHIP method essentially reduces the assumptions of the center path-following method and can be applied to more general problems.

Internal variable formulations of elastic-plastic dynamic problems

The paper reviews work that has been carried out in recent years in the Centre for Research in Computational and Applied Mechanics (CERECAM) at the University of Cape Town on the compact internal variable formulation of the problem of an elastic-plastic body subject to incremental loading. The fundamental problem is expressed as a convex nonlinear programming problem. The simplest two step algorithm for the solution of this programming problem is shown to be the standard Newton-Raphson algorithm, and the formulation permits a discussion of the convergence of the iterative solution procedure. It is shown that this static formulation is readily extended to include inertia and linear damping terms, leading to an identical basic formulation in which the stiffness matrix and the residual are redefined for the dynamic problem. The static problem, and hence any software written for its solution, can thus be simply modified to include dynamic effects. The essential analogy between the static and dynamic problems was first presented as a master's thesis (R.D. Isted, 1988); subsequently there have been a number of additional contributions to the basic formulation for static problems, in which the internal variable framework was extended and clarified. Further contributions have also been made in respect of the dynamic problem, and these developments are summarised. Some of this later work has been directed at improvements in the implicit solution algorithms. These aspects will not be discussed in detail, however, the main purpose of this review is to demonstrate the formal relationship between the static and dynamic problems.

Managing Uncertainty in a Serial Production Line

This paper studies a serial production line that is uncertain. We introduce a linear model of the uncertainty that can exist in the demand for the product and in each stage's processing time, yield, rework probability, and reliability. We construct a linear discrete-time rule for controlling production, and we show that repeated application of this rule leads the system to steady-state conditions, which include closed-form formulas for the mean and variance of each buffer's stock and the mean and variance of the workload in each stage. We optimize these operating characteristics by a convex nonlinear program. For the case of identical stages with no scrap, we show that this optimal solution tends to place larger buffer stocks toward the end of the line. We adapt these discrete-time results to the short-period surrogate of the analogous continuous-time control problem.

Newton's method for quadratic stochastic programs with recourse

Quadratic stochastic programs (QSP) with recourse can be formulated as nonlinear convex programming problems. By attaching a Lagrange multiplier vector to the nonlinear convex program, a QSP is written as a system of nonsmooth equations. A Newton-like method for solving the QSP is proposed and global convergence and local super-linear convergence of the method are established. The current method is more general than previous methods which were developed for box-diagonal and fully quadratic QSP. Numerical experiments are given to demonstrate the efficiency of the algorithm, and to compare the use of Monte-Carlo rules and lattice rules for multiple integration in the algorithm.

An extended cutting plane method for solving convex MINLP problems

An extended version of Kelley's cutting plane method is introduced in the present paper. The extended method can be applied for the solution of convex MINLP (mixed-integer non-linear programming) problems, while Kelley's cutting plane method was originally introduced for the solution of convex NLP (non-linear programming) problems only. The method is suitable for solving large convex MINLP problems with a moderate degree of nonlinearity. The convergence properties of the method are given in the present paper and an example is provided to illustrate the numerical procedure.

Stochastic user equilibrium assignment in networks with queues

A stochastic user equilibrium assignment algorithm is presented for steady state store-and-forward networks. The links of the network have constant travel times and the links or nodes have finite capacities. When capacity is reached, delay sufficient to match demand to the available capacity is generated. It has been shown by others that the equilibrium assignment in networks of this kind is the solution to a particular linear programming problem. By adding an entropy term to the objective function, a convex nonlinear programming problem is formed which yields a stochastic user equilibrium assignment. For the case of link constraints, it is proven that the Lagrange multipliers of both the linear and the non-linear programming problems give the equilibrium delays in the network. The requirements for uniqueness are investigated. Iterative algorithms are formulated for solving the nonlinear programming problem with either link or node constraints and convergence is proven. For networks where path enumeration is likely to be a problem, a column generation technique is proposed. An illustrative example is presented.

5388186 Differential process controller using artificial neural networks : Bose Chinmoy B Green Brook, NJ, UNITED STATES Assigned to AT&T Corp

An analytical point-wise stationary approximation model is proposed to analyze time-dependent truck queuing processes with stochastic service time distributions at gates and yards of a port terminal. A convex nonlinear programming model is developed which minimizes the total truck turn time and discomfort due to shifted arrival times. A two-phase optimization approach is used to first compute a system-optimal truck arrival pattern, and then find a desirable pattern of time-varying tolls that leads to the optimal arrival pattern. Numerical experiments are conducted to test the computational efficiency and accuracy of the proposed optimization models.

SHAKEDOWN OPTIMUM DESIGN OF REINFORCED CONCRETE FRAMED STRUCTURES

Structures subjected to variable repeated loads can undergo the shakedown or adaptation phenomenon,-which prevents them from collapse but may cause lack of serviceability, for the plastic deformations developed, although finite, as shakedown occurrence postulates, may exceed some maximum values imposed by external ductility criteria. This paper is devoted to the optimal design of reinforced concrete structures, subjected to variable and repeated loads. For such structures the knowledge of the actual values taken by the plastic deformations, at shakedown occurrence, is a crucial issue. An approximate assessment of such plastic deformations is needed, which is herein provided in the shape of quadratic constraints, by the so-called perturbation method. By suitable finite element discretization, the optimal shakedown problem is formulated as a convex nonlinear mathematical programming one. The M-N (bending moment-axial force) interaction is accounted for in the aim to provide the optimal shakedown design. The...

Analysis and design of trussed structures made of elastic/stiffening materials

An extremum problem formulation for the equilibrium analysis of general structures made of stiffening material is applied to the analysis of trussed structures. The nonlinear material is modelled in a way to simulate an arbitrary, polygonal stress-strain relation; material properties may vary over the truss. The form of this convex nonlinear programming problem statement is convenient for the prediction of the evolution (over the truss system) of local member stiffening under increasing proportional load. Computational solutions are obtained directly on the basis of the extremum problem statement, using commercial minimizer software. The formulation for analysis is extended to model a design problem for the prediction of the optimal modification of certain material properties, namely the bounds that reflect the onset of stiffening in the material. Computational results for this optimal material design problem are also provided.

Global optimization algorithm for heat exchanger networks

This paper deals with the global optimization of heat exchanger networks with fixed topology. It is shown that if linear area cost functions are assumed, as well as arithmetic mean driving force temperature differences in networks with isothermal mixing, the corresponding nonlinear programming (NLP) optimization problem involves linear constraints and a sum of linear fractional functions in the objective which are nonconvex. A rigorous algorithm is proposed that is based on a convex NLP underestimator that involves linear and nonlinear estimators for fractional and bilinear terms which provide a tight lower bound to the global optimum. This NLP problem is used within a spatial branch and bound method for which branching rules are given. Basic properties of the proposed method are presented, and its application is illustrated with several example problems. The results show that the proposed method only requires few nodes in the branch and bound search.

Linear Control of a Markov Production System

This paper presents a model of a facility that processes many jobs. Each job requires a sequence of activities. The sequence of activities required by a job is random because each activity ends with a test, and the outcome of that test determines the activity that must be performed next. Evolution from activity to activity is Markovian, i.e., is determined by transition probabilities. A few (e.g., six) different types of job can share this facility, each type having its own transition probabilities between the activities. Each activity takes place in a designated sector. Several activities can share a sector. Each sector has a capacity, and buffers of inventory serve to decouple the sectors. In this paper, we introduce a family of linear control rules that smooth the flow through each sector and regulate the amount of inventory in each buffer. The operating characteristics of each linear control rule are computed. These include the mean and variance of the flow into each sector, the inventory in each buffer, the throughput of the systems, and the cycle time. These operating characteristics are optimized by a convex nonlinear program.

COMPLEMENTARITY SYSTEMS AND OPTIMIZATION PROBLEMS IN STRUCTURAL ENGINEERING

In the late 1960s complementarity systems have been recognized to be natural models for some problems in structural mechanics. Complementarity systems are known to arise also as stationarity conditions in optimization problems. This paper, after a short survey of complementarity systems, their connections with variational inequalities and solution algorithms, deals with some recent nonconventional applications in structural engineering. Precisely, a fairly detailed description is provided of the optimum design problem of finding the minimum cost modifications of the supporting profile along a submarine pipeline. In this engineering optimization problem a complementary system is included among the constraints. Mathematically similar problems arise in the area of sluctural identification and optimization. All these problems are related to analysis problems (unilateral contact, elastoplasticity) which are amenable to complementarity systems or non-linear convex (often quadratic) programming. The present purp...

On some efficient interior point methods for nonlinear convex programming

We introduce two interior point algorithms for minimizing a convex function subject to linear constraints. Our algorithms require the solution of a nonlinear system of equations at each step. We show that if sufficiently good approximations to the solutions of the nonlinear systems can be found, then the primal-dual gap becomes less that ε in O( n |lnε|) steps, where n is the number of variables.

Consistent predictors and the solution of the piecewise holonomic incremental problem in elasto-plasticity

Algorithms for the solution of the classical incremental problems in static elasto-plasticity are an important feature in large-scale finite element programs for nonlinear stress analysis. The algorithms are based on the Newton-Raphson method, where a linearized iteration is used. In each iteration a predictor step is employed to compute a new, improved estimate of the kinematic variables, and a corrector step recomputes the stresses and the internal forces associated with the improved solution.Backward difference methods have been shown empirically to provide an effective and a stable means of implementing the corrector step. Very recently it has also been shown that so-called consistent predictors should be used together with the backward difference correctors in order to ensure the quadratic rate of convergence normally expected with the Newton-Raphson method. Very considerable success has been achieved with the use of consistent predictors, both in terms of stability and rate of convergence.In this paper we present a rational formulation of the incremental problem. It is shown that the adoption of a backward difference corrector is equivalent to the reformulation of the problem as a piecewise holonomic problem. This permits the problem to be written as an unconstrained convex nonlinear programming problem. It is further shown that the consistent predictor can easily be identified within this formulation, and as a consequence it can be interpreted as a rational step in the algorithm for the solution of the nonlinear programming problem.

Optimal module sizing in VLSI floorplanning by nonlinear programming

Floorplanning is the VLSI design problem of deciding the position and shapes of all modules (e.g., memory or random logic) in order to minimize the chip area and satisfy all constraints. The modules may not be fixed in shape — the ratio of height to width may be modified in order to achieve minimal chip area. In the CADRE (ATT the IBM system uses a Simplex-method based heuristic. These heuristics are not guaranteed to solve the problem exactly. Consequently, it is not known how far the heuristic solution is from optimal, and it is not obvious when to terminate the heuristic. In this paper we show that, given the relative positions, we can compute theprovably optimal shape of each module, by transforming the problem into a convex nonlinear programming problem. Each local minimum of such a problem must also be a global minimum. We illustrate the method by applying it to an example solved by IBM.

Branch and Bound Experiments in Convex Nonlinear Integer Programming

The branch and bound principle has long been established as an effective computational tool for solving mixed integer linear programming problems. This paper investigates the computational feasibility of branch and bound methods in solving convex nonlinear integer programming problems. The efficiency of a branch and bound method often depends on the rules used for selecting the branching variables and branching nodes. Among others, the concepts of pseudo-costs and estimations are implemented in selecting these parameters. Since the efficiency of the algorithm also depends on how fast an upper bound on the objective minimum is attained, heuristic rules are developed to locate an integer feasible solution to provide an upper bound. The different criteria for selecting branching variables, branching nodes, and heuristics form a total of 27 branch and bound strategies. These strategies are computationally tested and the empirical results are presented.

Improved Efficiency of the Frank-Wolfe Algorithm for Convex Network Programs

We discuss methods for speeding up convergence of the Frank-Wolfe algorithm for solving nonlinear convex programs. Models involving hydraulic networks, road networks and factory-warehouse networks are described. The PARTAN technique and heuristic variations of the Frank-Wolfe algorithm are described which serve to significantly improve the convergence rate with no significant increase in memory requirements. Computational results for large-scale models are reported.