Error analysis for convex separable programs: Bounds on optimal and dual optimal solutions

  title={Error analysis for convex separable programs: Bounds on optimal and dual optimal solutions},
  author={Lakshman S. Thakur},
  journal={Journal of Mathematical Analysis and Applications},
  • Lakshman S. Thakur
  • Published 1980
  • Mathematics
  • Journal of Mathematical Analysis and Applications
Abstract Computable lower and upper bounds on the optimal and dual optimal solutions of a nonlinear, convex separable program are obtained from its piecewise linear approximation. They provide traditional error and sensitivity measures and are shown to be attainable for some problems. In addition, the bounds on the solution can be used to develop an efficient solution approach for such programs, and the dual bounds enable us to determine a subdivision interval which insures the objective… Expand
Successive approximation in separable programming: an improved procedure for convex separable programs
We implement a solution procedure for general convex separable programs where a series of relatively small piecewise linear programs are solved as opposed to a single large one, and where, based onExpand
Domain Contraction in Nonlinear Programming: Minimizing a Quadratic Concave Objective Over a Polyhedron
Using linear underestimating approximations and their dual solutions, a combination of the proposed domain contraction via linear underestimation and its dual information with the branch and bound approach offers a computational strategy competitive with the currently dominant methods. Expand
Optimal objective function approximation for separable convex quadratic programming
The method provides guidelines on how many grid points to use and how to position them for a piecewise-linear approximation if the error induced by the approximation is to be bounded a priori. Expand
Sequence of polyhedral relaxations for nonlinear univariate functions
The letter develops a sequence of Mixed Integer Linear Programming (MILP) and Linear Programming (LP) relaxations that converge to the graph of a nonlinear, univariate, bounded, and differentiableExpand
Solving highly nonlinear convex separable programs using successive approximation
This paper demonstrates the extreme degree of vulnerability of standard separable programming to high nonlinearity, then states the algorithm and some of its important characteristics, and shows its effectiveness for computational examples. Expand
A MILP formulation for generalized geometric programming using piecewise-linear approximations
This approach is to approximate a multiple-term log-sum function of the form log in terms of a set of linear equalities or inequalities of log x1, log x2, …, and log xn, where x1 , …, xn are strictly positive. Expand
Constrained optimization in L∞-norm: An algorithm for convex quadratic interpolation
Abstract A geometrically convergent algorithm is presented to determine min ∥ƒ(2)∥∞, where ƒ interpolates the given p points {(xi, yi)}1p with increasing x i ' s , ƒ ϵ C 1 , ƒ is absolutelyExpand
A direct algorithm for optimal quadratic splines
SummaryWe develop an algorithm to findk, the minimal value of ‖f(2)‖∞, wheref∈C1 is a quadratic spline with free knots, which interpolates the givenp points {xi,yi}1p with increasingxi's, hasExpand


Error Analysis for Convex Separable Programs: The Piecewise Linear Approximation and The Bounds on The Optimal Objective Value
The bounds have been established on the possible deviation of the optimal objective value of a separable, convex program from the optimal objective value of a program which is its piecewise linearExpand
Programming Under Uncertainty: The Equivalent Convex Program
This paper is an attempt to describe and characterize the equivalent convex program of a two-stage linear program under uncertainty. The study has been divided into two parts. In the first one, weExpand
Objective function approximations in mathematical programming
It is shown that this criterion for selecting the “best” approximation from any given class is equivalent for all practical purposes to the familiar Chebyshev approximation criterion. Expand
On approximate solutions of systems of linear inequalities
(briefly, -4x^6), one arrives at a vector JC that "almost" satisfies (1). It is almost obvious geometrically that, if (1) is consistent, one can infer that there is a solution x0 of (1) "close" to x.Expand
Stability Theory for Systems of Inequalities. Part I: Linear Systems
This paper deals with the stability of systems of linear inequalities in partially ordered Banach spaces when the data are subjected to small perturbations. We show that a certain condition isExpand
Chance-Constrained Equivalents of Some Stochastic Programming Problems
This paper shows that chance-constrained equivalents exist for several stochastic programming problems that are concerned with selecting a decision vector which will optimize the expectation of aExpand
Deterministic Solutions for a Class of Chance-Constrained Programming Problems
A certainty equivalent model without random variables for the chance-constrained model is developed such that feasible and optimal solutions of a chance- Constrained problem and of its associated certainty equivalent problem coincide. Expand
On Perturbations in Systems of Linear Inequalities
We consider what happens to sets defined by systems of linear inequalities when elements of the system are perturbed. If $S = \{ x|Gx \leqq g,Dx = d\} $, and if ${S'}$ and ${S''}$ are defined in theExpand
Mathematical programming in practice
As one of the part of book categories, mathematical programming in practice always becomes the most wanted book. Expand