Showing papers on "Convex optimization published in 1989"

Direct Methods in the Calculus of Variations

Abstract: Introduction.- Convex Analysis and the Scalar Case.- Convex Sets and Convex Functions.- Lower Semicontinuity and Existence Theorems.- The one Dimensional Case.- Quasiconvex Analysis and the Vectorial Case.- Polyconvex, Quasiconvex and Rank one Convex Functions.- Polyconvex, Quasiconvex and Rank one Convex Envelopes.- Polyconvex, Quasiconvex and Rank one Convex Sets.- Lower Semi Continuity and Existence Theorems in the Vectorial Case.- Relaxation and Non Convex Problems.- Relaxation Theorems.- Implicit Partial Differential Equations.- Existence of Minima for Non Quasiconvex Integrands.- Miscellaneous.- Function Spaces.- Singular Values.- Some Underdetermined Partial Differential Equations.- Extension of Lipschitz Functions on Banach Spaces.- Bibliography.- Index.- Notations.

Interior path following primal-dual algorithms. Part II: Convex quadratic programming

Abstract: We describe a primal-dual interior point algorithm for convex quadratic programming problems which requires a total of\(O\left( {\sqrt n L} \right)\) number of iterations, whereL is the input size. Each iteration updates a penalty parameter and finds an approximate Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem. The algorithm is based on the path following idea. The total number of arithmetic operations is shown to be of the order of O(n3L).

Structured and Simultaneous Lyapunov Functions for System Stability Problems

Abstract: It is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic Lyapunov function of a certain structure which establishes stability of x = Ax for some appropriate A. The existence of such a Lyapunov function can be determined by solving a convex program. We present several numerical methods for these optimization problems. A simple numerical example is given. Proofs can be found in Boyd and Yang [BY88].

CONLIN: An efficient dual optimizer based on convex approximation concepts

Abstract: The Convex Linearization method (CONLIN) exhibits many interesting features and it is applicable to a broad class of structural optimization problems. The method employs mixed design variables (either direct or reciprocal) in order to get first order, conservative approximations to the objective function and to the constraints. The primary optimization problem is therefore replaced with a sequence of explicit approximate problems having a simple algebraic structure. The explicit subproblems are convex and separable, and they can be solved efficiently by using a dual method approach. In this paper, a special purpose dual optimizer is proposed to solve the explicit subproblem generated by the CONLIN strategy. The maximum of the dual function is sought in a sequence of dual subspaces of variable dimensionality. The primary dual problem is itself replaced with a sequence of approximate quadratic subproblems with non-negativity constraints on the dual variables. Because each quadratic subproblem is restricted to the current subspace of non zero dual variables, its dimensionality is usually reasonably small. Clearly, the Hessian matrix does not need to be inverted (it can in fact be singular), and no line search process is necessary. An important advantage of the proposed maximization method lies in the fact that most of the computational effort in the iterative process is performed with reduced sets of primal variables and dual variables. Furthermore, an appropriate active set strategy has been devised, that yields a highly reliable dual optimizer.

An extension of Karmarkar projective algorithm for convex quadratic programming

Abstract: We present an extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convex quadratic programs. This extension is based on the iterated application of the objective augmentation and the projective transformation, followed by optimization over an inscribing ellipsoid centered at the current solution. It creates a sequence of interior feasible points that converge to the optimal feasible solution in O(Ln) iterations; each iteration can be computed in O(Ln3) arithmetic operations, wheren is the number of variables andL is the number of bits in the input. In this paper, we emphasize its convergence property, practical efficiency, and relation to the ellipsoid method.

From Convex Optimization to Nonconvex Optimization. Necessary and Sufficient Conditions for Global Optimality

Abstract: Nonconvex minimization problems form an old subject which has received a growing interest in the recent years. The main incentive comes from modelling in Applied Mathematics and Operations Research, where one may be faced with optimization problems like: minimizing (globally) a difference of convex functions, maximizing a convex function over a convex set, minimizing an indefinite quadratic form over a polyhedral convex set, etc.

A Direct Minimization Approach for Obtaining the Distance between Convex Polyhedra

Abstract: Computational methods used for the automatic generation of robot paths must be fully developed if truly automated manu facturing systems are to become a reality. An important requirement for determining feasible robot paths is the ability to compute the distance between the various elements of the robot and the workspace fixtures, jigs, and machinery. In this research, it is assumed that the robot and workspace solid geometry are represented as a collection of convex polyhe drons, and an efficient numerical algorithm for determining the minimum distance between two such polyhedrons is presented. In addition to determining the minimum distance between solids, the algorithm can also be used to efficiently ascertain whether a collision has occurred.The numerical technique presented uses a sequence of constrained minimizations to obtain the closest three-dimen sional points on any two solid objects. Computational effi ciency is achieved with the algorithm presented because only the collection of planes (point...

Tradeoff Curves, Targeting and Balancing in Manufacturing Queueing Networks

Abstract: In this paper, we introduce the notions of tradeoff curves, targeting and balancing in manufacturing systems to describe the relationship between variables such as work-in-process, lead-time and capacity. We consider multiproduct manufacturing systems modeled by open networks of queues and formulate the targeting TP and balancing BP problems as nonlinear programs. These formulations are based primarily on parametric decomposition methods for estimating performance measures in open queueing networks. Since TP and BP typically are hard to solve, we show that under fairly realistic conditions they can be approximated by easily solvable convex programs. We present heuristics to obtain approximate solutions to these problems and to derive tradeoff curves. We also provide bounds on the performance of the heuristics, relative to the approximation problems, and show that they are asymptotically optimal under mild conditions.

Pure adaptive search in Monte Carlo optimization

Abstract: Pure adaptive search constructs a sequence of points uniformly distributed within a corresponding sequence of nested regions of the feasible space. At any stage, the next point in the sequence is chosen uniformly distributed over the region of feasible space containing all points that are equal or superior in value to the previous points in the sequence. We show that for convex programs the number of iterations required to achieve a given accuracy of solution increases at most linearly in the dimension of the problem. This compares to exponential growth in iterations required for pure random search.

Convex duality approach to the optimal control of diffusions

Abstract: An alternative to the usual dynamic programming approach to the optimal control of Markov processes is considered. It is based on duality of convex analysis. The control problem is embedded in a convex mathematical programming problem on a space of measures. The dual problem is to find the supremum of smooth subsolutions to the Hamilton–Jacobi–Bellman equation.

Efficient approximation concepts using second order information

Abstract: The primary goal of this paper is to show how second derivative information can be used in an effective way in structural optimization problems. The basic idea is to generate such an information at the expense of only one more ‘virtual load case’ in the sensitivity analysis part of the finite element code. To achieve this goal a primal–dual approach is employed, that can also be interpreted as a sequential quadratic programming method. Another objective is to relate the proposed method to the well known family of approximation concepts techniques, where the primary optimization problem is transformed into a sequence of non-linear explicit subproblems. When restricted to diagonal second derivatives, the new approach can be viewed as a recursive convex programming method, similar to the ‘Convex Linearization’ method (CONLIN), and to its recent generalization, the ‘Method of Moving Asymptotes’ (MMA). This new method has been successfully tested on simple problems that can be solved in closed form, as well as on sizing optimization of trusses. In all cases the method converges faster than CONLIN, MMA or other approximation techniques based on reciprocal variables.

The complexity of cutting complexes

Abstract: This paper investigates the combinatorial and computational aspects of certain extremal geometric problems in two and three dimensions. Specifically, we examine the problem of intersecting a convex subdivision with a line in order to maximize the number of intersections. A similar problem is to maximize the number of intersected facets in a cross-section of a three-dimensional convex polytope. Related problems concern maximum chains in certain families of posets defined over the regions of a convex subdivision. In most cases we are able to prove sharp bounds on the asymptotic behavior of the corresponding extremal functions. We also describe polynomial algorithms for all the problems discussed.

A primal-dual conjugate subgradient algorithm for specially structured linear and convex programming problems

Abstract: This paper presents a primal-dual conjugate subgradient algorithm for solving convex programming problems. The motivation, however, is to employ it for solving specially structured or decomposable linear programming problems. The algorithm coordinates a primal penalty function and a Lagrangian dual function, in order to generate a (geometrically) convergent sequence of primal and dual iterates. Several refinements are discussed to improve the performance of the algorithm. These are tested on some network problems, with side constraints and variables, faced by the Freight Equipment Management Program of the Association of American Railroads, and suggestions are made for implementation.

Optimization of globally convex functions

Abstract: Convex functions have nice properties with respect to both minimization and maximization. Similar properties are established here for functions that are permitted to have bad local behavior but are globally convex in the sense that they behave “convexly” on triples of collinear points that are widely dispersed. The results illustrate a development that seems desirable in the interest of more realistic mathematical modeling: the “globalization” of important function properties. In connection with the maximization of globally convex functions over convex bodies in a given finite-dimensional normed space E, there is interest in estimating the maximum, for points c of bodies $C \subset E$, of the ratio between two measures of how close c comes to being an extreme point of C. Good estimates are obtained for the cases in which E is Euclidean or has the “max” norm.

The projective SUMT method for convex programming

Abstract: An algorithm for solving convex programming problems is derived from the differential equation characterizing the trajectory of unconstrained minimizers of the classical logarithmic barrier function. Convergence of the continuous version of this projective SUMT method is proved under minimal assumptions. Extension of the algorithm to a form which handles linear equality constraints produces a differential equation analogue of Karmarkar's method for linear programming. The discrete version uses the same method of search and finds the step size by minimizing the logarithmic method of centers function. An acceleration procedure based on the nonvanishing of the Jacobian of the Karush-Kuhn-Tucker system at a minimizer is shown to converge quadratically. When the problem variables are bounded, dual feasible points are available and the algorithm produces at each iteration lower and upper bounds on the global minimum. A matrix approximation is given which greatly reduces the traditional problems in inverting the...

A Dual Method for Certain Positive Semidefinite Quadratic Programming Problems

Abstract: This paper presents a dual active set method for minimizing a sum of piecewise linear functions and a strictly convex quadratic function, subject to linear constraints. It may be used for direction finding in nondifferentiable optimization algorithms and for solving exact penalty formulations of (possibly inconsistent) strictly convex quadratic programming problems. An efficient implementation is described extending the Goldfarb and Idnani algorithm, which includes Powell's refinements. Numerical results indicate excellent accuracy of the implementation.

On the method of analytic centers for solving smooth convex programs

Abstract: We give a complexity analysis concerning the global convergence of the method of analytic centers for solving generalized smooth convex programs. We prove that the analytic center of the feasible set provides a two-sided ellipsoidal approximation of this set, whose tightness, as well as the global rate of convergence of the algorithm, only depends on the number of constraints and on a relative Lipschitz constant of the Hessian matrices of the constraint functions, but not on the data of the constraint functions. This work extends the results in [5] where the solution of problems with convex quadratic constraint functions has been discussed.

Duality algorithms for nonconvex variational principles

Abstract: This paper describes, and analyzes, a method of successive approximations for finding critical points of a function which can be written as the difference of two convex functions. The method is based on using a non-convex duality theory. At each iteration one solves a convex, optimization problem. This alternates between the primal and the dual variables. Under very general structural conditions on the problem, we prove that the resulting sequence is a descent sequence, which converges to a critical point of the problem. To illustrate the method, it is applied to some weighted eigenvalue problems, to a problem from astrophysics, and to some semilinear elliptic equations.

Optimal routing in circuit switched communication networks

Abstract: Consideration is given to the optimal circuit routing problem in an existing circuit-switched network. The objective is to find circuit routing which accommodates a given circuit demand while maximizing the residual capacity of the network. In addition, the cost of accommodating the circuit demand should not exceed a given amount. Practical considerations require that a solution be robust to the variations in circuit demand and cost. The objective function for the optimal circuit routing problem is not a smooth one. In order to overcome the difficulties of nonsmooth optimization, the objective function is approximated by smooth concave functions. The optimization algorithm for the circuit routing problem is obtained as a limiting case of the sequence of optimal routing strategies for the corresponding smooth convex optimization problems, and the proof of its convergence to the optimal solution is given. An approach to calculating the optimal multicommodity flow is presented. The optimization algorithm efficiently handles networks with a large number of commodities, satisfies the robustness requirements, and can be used to solve circuit routing problems for large networks. >

Theory of Duality in Mathematical Programming

Abstract: Contents: Convex sets and convex functions.- Theory of duality.- Dual problems of optimization.- Literature.- Notation Index.- Subject Index.

An ellipsoid trust region bundle method for nonsmooth convex minimization

Abstract: This paper presents a bundle method of descent for minimizing a convex (possibly nonsmooth) function f of several variables. At each iteration the algorithm finds a trial point by minimizing a polyhedral model of f subject to an ellipsoid trust region constraint. The quadratic matrix of the constraint, which is updated as in the ellipsoid method, is intended to serve as a generalized “Hessian” to account for “second-order” effects, thus enabling faster convergence. The interpretation of generalized Hessians is largely heuristic, since so far this notion has been made precise by J. L. Goffin only in the solution of linear inequalities. Global convergence of the method is established and numerical results are given.

Exact penalty functions for nondifferentiable programming problems

Abstract: In recent years an increasing attention has been devoted to the use of nondifferentiable exact penalty functions for the solution of nonlinear programming problems. However, as pointed out in [22], virtually all the published literature on exact penalty functions treats one of two cases: either the nonlinear programming problem is a convex problem (see, e.g., [2], [18], [23]), or it is a smooth problem (see, e.g., [1], [3–5], [10–13], [16], [18–20]). Exact penalty functions for nonlinear programming problems neither convex nor smooth, have been considered in [6], [21], [22], where locally lipschitz problems are dealt with.