A Lagrangian treatment of certain nonlinear clustering problems

more constraints or equations, with either a linear or nonlinear objective function. This distinction is made primarily on the basis of the difficulty of solving these two types of nonlinear problems. The first type is the less difficult of the two, and in this, Part I of the paper, it is shown how it is solved by the gradient projection method. It should be noted that since a linear objective function is a special case of a nonlinear objective function, the gradient projection method will also solve a linear programming problem. In Part II of the paper [16], the extension of the gradient projection method to the more difficult problem of nonlinear constraints and equations will be described. The basic paper on linear programming is the paper by Dantzig [5] in which the simplex method for solving the linear programming problem is presented. The nonlinear programming problem is formulated and a necessary and sufficient condition for a constrained maximum is given in terms of an equivalent saddle value problem in the paper by Kuhn and Tucker [10]. Further developments motivated by this paper, including a computational procedure, have been published recently [1]. The gradient projection method was originally presented to the American Mathematical Society

This paper presents a method, called the convex simplex method, for minimizing a convex objective function subject to linear inequality constraints. The method is a true generalization of Dantzig's linear simplex method both in spirit and in the fact that the same tableau and variable selection techniques are used. With a linear objective function the convex simplex method reduces to the linear simplex method. Moreover, the convex simplex method actually behaves like the linear simplex method whenever it encounters a linear portion of a convex objective function. Many of the sophisticated techniques designed to enhance the efficiency of the linear simplex method are applicable to the convex simplex method. In particular, as an example, a network transportation problem with a convex objective function is solved by using the standard transportation tableau and by only slightly modifying the usual procedure for a linear objective function.

The convex hull heuristic is a heuristic for mixed-integer programming problems with a nonlinear objective function and linear constraints. It is a matheuristic in two ways: it is based on the mathematical programming algorithm called simplicial decomposition, or SD (von Hohenbalken in Math Program 13:49–68, 1977), and at each iteration, one solves a mixed-integer programming problem with a linear objective function and the original constraints, and a continuous problem with a nonlinear objective function and a single linear constraint. Its purpose is to produce quickly feasible and often near optimal or optimal solutions for convex and nonconvex problems. It is usually multi-start. We have tested it on a number of hard quadratic 0–1 optimization problems and present numerical results for generalized quadratic assignment problems, cross-dock door assignment problems, quadratic assignment problems and quadratic knapsack problems. We compare solution quality and solution times with results from the literature, when possible.

An effective genetic algorithm approach to the quadratic minimum spanning tree problem

Portfolio optimization problems are rather easy to solve if one assumes normality of the (joint) distribution of returns with given parameters and a linear objective function, subject to some risk constraints. Higher degrees of complexities arise when taking into account (i) non- linear asset classes, (ii) non-normal distributions, (iii) constraints on lot size, portfolio shares etc., and (iv) uncertainty of parameter estimates. In this paper a Reliability-Based Design Optimization (RBDO) framework is developed to tackle the last two issues, namely, to solve portfolio optimization problems, taking into account the constraints on lot sizes, portfolio shares, etc., and the uncertainty of parameter estimates (i.e., return and risk). Thereby, standard methods and heuristics, available in the literature for handling reliability-based constraints, are adopted. The Block-Bootstrap method is used to obtain estimates of the uncertainty of parameter estimates. For an application to stocks which are components of the DAX, efficient portfolios are obtained taking into account the trade-off between optimal values of the objective function and the reliability of results.

A modified version of the weighted least absolute value (WLAV) method for the solution of the power system-state-estimation problem is presented. The WLAV method can be seen as minimising a linear objective function subject to a set of nonlinear constraints. The modification is aimed at producing a WLAV estimator that remains insensitive to bad data, even if they are associated with leverage-point measurements. This is achieved by bending the linear objective function, so that the residuals of bad measurements are allowed to grow without incurring much additional cost in the objective function. Consequently, the optimisation procedure would find an optimal solution where the residuals of all bad measurements are nonzero. The optimisation is carried out via sequential linear programming. It is shown that each linear program corresponds to a linearised WLAV problem with weights adjusted automatically during the iterations. To ensure fast execution time, the linear program is solved using a homogeneous interior-point method. Computational results show that the proposed method can identify bad data in leverage points.

A model for optimal operation of water supply/irrigation systems of various water quality sources, with treatment plants, multiple water quality conservative factors, and dilution junctions is presented. The objective function includes water cost at the sources, water conveyance costs which account for the hydraulics of the network indirectly, water treatment cost, and yield reduction costs of irrigated crops due to irrigation with poor quality water. The model can be used for systems with supply by canals as well as pipes, which serve both drinking water demands of urban/rural consumers and field irrigation requirements. The general nonlinear optimization problem has been simplified by decomposing it to a problem with linear constraints and nonlinear objective function. This problem is solved using the projected gradient method. The method is demonstrated for a regional water supply system in southern Israel that contains 39 pipes, 37 nodes, 11 sources, 10 agricultural consumers, and 4 domestic consumers. The optimal operation solution is described by discharge and salinity values for all pipes of the network. Sensitivity of the optimal solution to changes in the parameters is examined. The solution was found to be sensitive to the upper limit on drinking water quality, with total cost being reduced by 5% as the upper limit increases from 260 to 600 mg Cl l−1. The effect of income from unit crop yield is more pronounced. An increase of income by a factor of 20 results in an increase of the total cost by a factor of 3, thus encouraging more use of fresh water as long as the marginal cost of water supply is smaller than the marginal decrease in yield loss. The effect of conveyance cost becomes more pronounced as its cost increases. An increase by a factor of 100 results in an increase of the total cost by about 14%. The network studied has a long pipe that connects two distinct parts of the network and permits the supply of fresh water from one part to the other. Increasing the maximum permitted discharge in this pipe from 0 to 200 m3 h−1 reduces the total cost by 11%. Increasing the maximum discharge at one of the sources from 90 to 300 m3 h−1 reduces the total cost by about 8%.

In this paper, a new algorithm to solve a general 0–1 programming problem with linear objective function is developed. Computational experiences are carried out on problems where the constraints are inequalities on polynomials. The solution of the original problem is equivalent with the solution of a sequence of set packing problems with special constraint sets. The solution of these set packing problems is equivalent with the ordering of the binary vectors according to their objective function value. An algorithm is developed to generate this order in a dynamic way. The main tool of the algorithm is a tree which represents the desired order of the generated binary vectors. The method can be applied to the multi-knapsack type nonlinear 0–1 programming problem. Large problems of this type up to 500 variables have been solved.

Straightness and flatness tolerance evaluation: an optimization approach

Chapter two - Linear programming

This issue's SIGEST paper, from the SIAM Journal on Optimization (SIOPT), is an excellent example of deep mathematical analysis, involving a range of mathematical techniques, which is having significant practical algorithmic and computational consequences. The title of the paper, “A Sum of Squares Approximation of Nonnegative Polynomials,” by Jean B. Lasserre, combines two concepts that are known to all SIAM readers: sums of squares, the types of functions that arise in many data fitting and other applications; and nonnegative polynomials, polynomials whose values are greater than or equal to zero for all inputs. It is obvious that any polynomial that can be written as a sum of squares is nonnegative. It is far from obvious whether any nonnegative polynomial has a representation as, or arbitrarily close to, a sum of squares of polynomials. In this SIGEST paper, Lasserre shows that this is the case. The nomination from the editors of SIOPT captures the immediate and more general contributions of this paper eloquently, in the context of a series of three papers of which this is the second: “Jean Lasserre, in a series of papers (starting from the first paper in SIOPT Vol. 11, pp. 796–817, and concluding with the third paper in SIOPT Vol. 15, pp. 383–393) by exploiting deep results from real algebraic geometry and theory of moments for multivariate polynomials, showed that the optimal value of a nonconvex polynomial optimization problem (i.e., an optimization problem whose objective function and constraints are specified by multivariate polynomials) can asymptotically be approximated by an increasing sequence of optimal values from increasing tighter semidefinite programming relaxation problems. Even though the result is asymptotic in the most general case, fortuitously, practical experience has revealed that very often, the first few levels of semidefinite programming relaxations typically can achieve the global optimum value. The theoretical connection that Jean Lasserre (and Pablo Parrilo at about the same time from the viewpoint of sum of squares representations of multivariate polynomials) had established between polynomial optimizations with real algebraic geometry and theory of moments has led to exciting developments in combinatorial optimization. In particular, many hard combinatorial problems such as the maximum clique problem of a graph and the quadratic assignment problem are now actively revisited by researchers under the framework of polynomial optimization to derive increasingly tighter bounds.” That is, a consequence of this and related work is the ability to use a sequence of semidefinite programming problems to (nearly) solve much more difficult optimization problems. Semidefinite programming problems are problems with a linear objective function and linear constraints in a matrix variable X, plus the additional constraint that X be positive semidefinite. Since a rich array of efficient algorithms for solving semidefinite programming problems has been developed in recent years, reducing more general problems to a sequence of semidefinite programming problems is an important advance. We hope that readers of SIAM Review will enjoy this glimpse into the sophisticated mathematics behind an important algorithmic innovation in optimization.

Dynamics and steady state analysis of controlled Generalized Batches Petri Nets

This article introduces a Hybrid Intervention Scheme Based Optimization (HIBO) algorithm solving an Optimal Reactive Power Management (ORPM) problem in real-time using a Mixed Integer Linear Programming (MILP) solver. The ORPM problem presented here contains a linear objective function containing four objectives separated using a set of static penalty factors for each area. The non-linear optimization problem has been assumed linear by localizing the search for solution, this is done by introducing a penalty on the change from the original state or the base case scenario. Thereby, optimizing the non-linear ORPM in linear steps makes it a fast solver for small changes in power system state. A contingency analysis (for N-1 voltage violations) is included for ensuring the safety and reliability of the power system. The results are used to update the ORPM problem or stop if the system is secure. The optimization variables used to represent transformer taps and shunt device switches are handled as discrete integers and remaining variables as continuous real numbers. The intervention scheme, objectives and constraints used in the HIBO have been derived through surveys conducted at a transmission system control center and are supported using literature. Validation of the HIBO algorithm was performed on the Dutch transmission network model after dividing it into four regional areas. Convergence characteristics of the HIBO algorithm are compared using results. From the results, it is concluded that the HIBO algorithm is a competitive optimization solver, suitable for deployment in the secondary voltage control scheme within system operations domain for transmission system operators.

In this paper, we propose a method for estimating the algebraic Riccati equation (ARE) with respect to an unknown discrete-time system from the system state and input observation. The inverse optimal control (IOC) problem asks, “What objective function is optimized by a given control system?” The inverse linear quadratic regulator (ILQR) problem is an IOC problem that assumes a linear system and quadratic objective function. The ILQR problem can be solved by solving a linear matrix inequality that contains the ARE. However, the system model is required to obtain the ARE, and it is often unknown in fields in which the IOC problem occurs, for example, biological system analysis. Our method directly estimates the ARE from the observation data without identifying the system. This feature enables us to economize the observation data using prior information about the objective function. We provide a data condition that is sufficient for our method to estimate the ARE. We conducted a numerical experiment to demonstrate that our method can estimate the ARE with less data than system identification if the prior information is sufficient.

Algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in the case of variational inequalities. In this paper, we derive sufficient conditions for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for variational inequalities.