# A simplex-type algorithm for continuous linear programs with constant coefficients

## TL;DR

This paper introduces a simplex-type algorithm for solving continuous linear programs with constant coefficients over a finite time horizon, extending classical LP concepts to measure spaces and ensuring finite-step optimal solutions.

## Contribution

It develops a finite-step simplex-like algorithm for continuous linear programs in measure spaces, generalizing previous models and their duals.

## Key findings

- Algorithm solves problems in finite steps
- Extends LP theory to measure and function spaces
- Ensures existence of optimal strongly dual solutions

## Abstract

We consider continuous linear programs over a continuous finite time horizon $T$, with a constant coefficient matrix, linear right hand side functions and linear cost coefficient functions, where we search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the separated continuous linear programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. In previous papers we have shown that these problems possess optimal strongly dual solutions. We also have presented a detailed description of optimal solutions and have defined a combinatorial analogue to basic solutions of standard LP. In this paper we present an algorithm which solves this class of problems in a finite bounded number of steps, using an analogue of the simplex method, in the space of measures.