The method of feasible directions for mathematical programming problems with preconvex constraints

The convergence of the method of feasible directions is proved for the case of the smooth objective function and a constraint in the form of the difference of convex sets (the so-called preconvex set). It is shown that the method converges to the set of stationary points, which generally is narrower than the corresponding set in the case of a smooth function and smooth constraints. The scheme of the proof is similar to that proposed earlier by Karmanov.     

Minimax linear programming problem

In this paper, we consider the following minimax linear programming problem: min z = max_{1 ≤ j ≤ n}{C_jX_j}, subject to Ax = g, x ≥ 0. It is well known that this problem can be transformed into a linear program by introducing n additional constraints. We note that these additional constraints can be considered implicitly by treating them as parametric upper bounds. Based on this approach we develop two algorithms: a parametric algorithm and a primal—dual algorithm. The parametric algorithm solves a linear programming problem with parametric upper bounds and the primal—dual algorithm solves a sequence of related dual feasible linear programming problems. Computation results are also presented, which indicate that both the algorithms are substantially faster than the simplex algorithm applied to the enlarged linear programming problem.     

An efficient algorithm for solving inequalities

An efficient algorithm for solving a finite system of inequalities in a finite number of iterations is described and analyzed.This work was supported by the UK Science and Engineering Research Council     

On the efficacy of pseudo-coordinates—Part 2: moving boundary constraints

In this paper an argument is presented in favor of utilizing felicitous or natural coordinates in the model formulation of complex hybrid parameter multiple body mechanical systems (HPMBS). Specifically for this paper, HPMBS that consist of continuua that are subjected to spatially and temporally varying non-holonomic boundary conditions. This is the second paper of a two part series of papers that is presented to clarify the novelty and usefulness of a recently developed Gibbs-Appell type projection based HPMBS modeling tool. The purpose of the paper is to show that with the novel use of pseudo-coordinates and speeds (as defined by the author) it is completely natural to provide minimal configuration space dimensionality yet still retain rigorous analytical formulation tractability.Presented in this work, as a demonstrative arguing point, is the development of the hybrid parameter motion equations for a rolling flexible-disk material cutting device. This device consists of a circular flexible continuum (the cutter) along with the requisite mounting rigid hub and handle. This non-holonomically constrained device is modeled executing spatial motion constrained to the plane via moving constraints applied to the boundary of the planar continuum. Also included in this work are numerical results bolstering the claims made herein. These numerical results demonstrate that the methodology elucidated provides low-order models suitable for modeling complicated devices. These low-order models are in contrast to the current modeling trend of ever-increasing degrees of freedom.     

Optimal control problems with state constraints for semilinear distributed-parameter systems

We consider optimal control problems for distributed-parameter systems described by semilinear equations, with constraints on the control and on the state, and an exact pointwise target condition. As an application of a general theory of nonlinear programming problems in Banach spaces, a version of the Pontryagin maximum principle is obtained.This research was partly supported by the National Science Foundation under Grant DMS-92-21819.     

EXISTENCE OF FORECAST AND DECISION HORIZONS FOR OPTIMAL CONTROL PROBLEMS OF NONLINEAR SYSTEMS

ＥＸＩＳＴＥＮＣＥＯＦＦＯＲＥＣＡＳＴＡＮＤＤＥＣＩＳＩＯＮＨＯＲＩＺＯＮＳＦＯＲＯＰＴＩＭＡＬＣＯＮＴＲＯＬＰＲＯＢＬＥＭＳＯＦＮＯＮＬＩＮＥＡＲＳＹＳＴＥＭＳ￥ＺｈａｎｇＣｈｅｎｇｙｕｎ（张成云）（ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，复旦大学，邮编：...     

Analytical solutions to the optimization of a quadratic cost function subject to linear and quadratic equality constraints

In the area of broad-band antenna array signal processing, the global minimum of a quadratic equality constrained quadratic cost minimization problem is often required. The problem posed is usually characterized by a large optimization space (around 50–90 tuples), a large number of linear equality constraints, and a few quadratic equality constraints each having very low rank quadratic constraint matrices. Two main difficulties arise in this class of problem. Firstly, the feasibility region is nonconvex and multiple local minima abound. This makes conventional numerical search techniques unattractive as they are unable to locate the global optimum consistently (unless a finite search area is specified). Secondly, the large optimization space makes the use of decision-method algorithms for the theory of the reals unattractive. This is because these algorithms involve the solution of the roots of univariate polynomials of order to the square of the optimization space. In this paper we present a new algorithm which exploits the structure of the constraints to reduce the optimization space to a more manageable size. The new algorithm relies on linear-algebra concepts, basic optimization theory, and a multivariate polynomial root-solving tool often used by decision-method algorithms.This research was supported by the Australian Research Council and the Corporative Research Centre for Broadband Telecommunications and Networking.     

An algorithm for maximizing entropy subject to simple bounds

An unusual form of the maximum entropy problem is considered, that includes simple bound constraints on the Fourier coefficients of the required image, as well as nonnegativity conditions on the image intensities. The algorithm avoids mixing these constraints by introducing a parameter into the objective function that is adjusted by an outer iteration. For each parameter value an inner iteration solves a large optimization calculation, whose constraints are just the simple bounds, by a combination of the conjugate gradient procedure and an active set method. An important feature is the ability to make many changes to the active set at once. The outer iteration includes a test for inconsistency of all the given constraints. The algorithm is described, a proof of convergence is given, and there are some second-hand remarks on numerical results.     

Linear programs with an additional rank two reverse convex constraint

An efficient algorithm is developed for solving linear programs with an additional reverse convex constraint having a special structure. Computational results are presented and discussed.     

Mixed constrained control problems

Necessary optimality conditions are derived in the form of a weak maximum principle for optimal control problems with mixed state-control equality and inequality constraints. In contrast to previous work these conditions hold when the Jacobian of the active constraints, with respect to the unconstrained control variable, has full rank. A feature of these conditions is that they are stated in terms of a joint Clarke subdifferential. Furthermore the use of the joint subdifferential gives sufficiency for nonsmooth, normal, linear convex problems. The main point of interest is not only the full rank condition assumption but also the nature of the analysis employed in this paper. A key element is the removal of the constraints and application of Ekeland's variational principle.