Piecewise Linear Cost Functions Research Articles

A Branch-and-Cut Algorithm Without Binary Variables for Nonconvex Piecewise Linear Optimization

We give a branch-and-cut algorithm for solving linear programs (LPs) with continuous separable piecewise-linear cost functions (PLFs). Models for PLFs use continuous variables in special-ordered sets of type 2 (SOS2). Traditionally, SOS2 constraints are enforced by introducing auxiliary binary variables and other linear constraints on them. Alternatively, we can enforce SOS2 constraints by branching on them, thus dispensing with auxiliary binary variables. We explore this approach further by studying the inequality description of the convex hull of the feasible set of LPs with PLFs in the space of the continuous variables, and using the new cuts in a branch-and-cut scheme without auxiliary binary variables. We give two families of valid inequalities. The first family is obtained by lifting the convexity constraints. The second family consists of lifted cover inequalities. Finally, we report computational results that demonstrate the effectiveness of our cuts, and that branch-and-cut without auxiliary binary variables is significantly more practical than the traditional mixed-integer programming approach.

Optimization in a vintage capital model with piecewise linear cost function

The paper investigates a nonlinear optimal control problem for the Volterra integral equations with an unknown function in the lower limit of integration. The applied interpretation of the problem consists of the optimal renovation of vintage capital under the embodied technological change. An explicit solution is constructed and its properties are analyzed in the case of a nonlinear revenue dependence on product output.

Efficient pricing of a bundled product of both real and reactive power

This paper examines a price mechanism with one price assigned for each level of bundled real and reactive power. Equilibrium allocation under this pricing approach raises system efficiency via better allocation of the reactive power reserves, neglected in the traditional pricing approach. Pricing reactive power separately is not very practical since its cost is highly dependent on real power output. Equilibrium allocation of the bundled pricing was simulated on the simple 3-bus system power auction and compared with free reactive power optimal power flow solution. The efficiency of this approach was shown in the general case, and tested on the 30-bus IEEE network with piecewise linear cost functions of the generators.

Models for representing piecewise linear cost functions

We study formulations of linear programs with piecewise linear objective functions with and without additional binary variables. We show that the two formulations without additional binary variables have the same LP bounds as those of the corresponding formulations with binary variables and therefore are preferable for efficient computation.

The Method for Solving a Piecewise-Linear Multicommodity Flow Problem

The paper deals with a flow distribution problem with a piecewise-linear cost function. The problem is formulated as a piecewise-linear programming problem which is not separable with respect to separate variable group. The method for solving this problem is based on the extension of the idea of the simplex method to the class of non-separable piecewise-linear problems. It secures finding of a local solution to the problem after a finite number of iterations. The method uses the peculiarities of the problem constraints that make it possible to decompose the matrix of constraints to smaller ones and thus to diminish the volume of calculations.

Optimal structural design via optimality criteria as a nonsmooth mechanics problem

In the theory of plastic structural design via optimality criteria (due to W. Prager), the optimal design problem is transformed to a nonlinear elastic structural analysis problem with appropriate stress-strain laws, which generally include complete vertical branches. In this context, the concept of structural universe (in the sense of G. Rozvany) permits the treatment of complicated optimal layout problems. Recent progress in the field of nonsmooth mechanics makes the solution of structural analysis problems with this kind of “complete” law possible. Elements from the two fields are combined in this paper for the solution of optimal design and layout problems for structures. The optimal layout of plane trusses with various specific cost functions is studied here as a representative problem. The use of convex, continuous and piecewise linear specific cost functions for the structural members leads to problems of linear variational inequalities or equivalently piecewise linear, convex but nonsmooth optimization problems, which are solved by means of an iterative algorithm based on sequential linear programming techniques. Numerical examples illustrate the theory and its applicability to practical engineering structures. Following a parametric investigation of an optimal bridge design, certain aspects of the optimal truss layout problem are discussed, which can be extended to other types of structural systems as well.

Short-term generation scheduling with transmission and environmental constraints using an augmented Lagrangian relaxation

This paper proposes a new approach based on augmented Lagrangian relaxation for short term generation scheduling problems with transmission and environmental constraints. In this method, the power system constraints, e.g. load demand, spinning reserve, transmission capacity and environmental constraints, are relaxed by using Lagrangian multipliers, and quadratic penalty terms associated with power system load demand balance are added to the Lagrangian objective function. Then, the decomposition and coordination technique is used, and nonseparable quadratic penalty terms are replaced by linearization around the solution obtained from the previous iteration. In order to improve the convergence property, the exactly convex quadratic terms of decision variables are added to the objective function as strongly convex, differentiable and separable auxiliary functions. The overall problem is decomposed into N subproblems, multipliers and penalty coefficients are updated in the dual problem and power system constraints are satisfied iteratively. The corresponding unit commitment subproblems are solved by dynamic programming, and the economic dispatch with transmission and environmental constraints is solved by an efficient network flow programming algorithm. The augmented Lagrangian relaxation method enhanced by the decomposition and coordination techniques avoids oscillations associated with piece-wise linear cost functions. Numerical results indicate that the proposed approach is fast and efficient in dealing with numerous power system constraints. >

Modelling piecewise linear concave costs in a tree partitioning problem

An important modelling question is that of how to obtain tight mixed integer programming formulations for problems involving piecewise linear concave cost functions. We examine in detail a tree partitioning model, and then specialise the results to the economic lot-sizing problem.

A direct simplex algorithm for network flow problems with convex piecewise linear costs

Minimum cost network flow problems with a piecewise linear convex cost function are used to model various optimization problems. They are also used extensively to approximate nonlinear cost functions which may otherwise be difficult to handle. Solving the piecewise linear problems using a reformulation approach is possible but may be inefficient. In thispaper we discuss a specialized direct approach and its implementation for solving such problems. The direct approach handles the piecewise linear structure of the cost function by allowing each arc to have varying costs on different segments. Computational results havel been reported, from which we conclude that using such an approach has adistinct advantage over using a reformulation approach.

Single-Machine Scheduling for Minimizing the Expected Total Cost Subject to Machine Breakdowns.

A single-machine scheduling model is proposed for minimizing the expected total cost consisting of flowtime cost and asymmetric penalty cost of earliness-tardiness subject to machine breakdowns. All jobs have a common due date and an identical piecewise linear cost function. Since it is difficult to calculate the exact value of expected total cost, an approximate function of expected total cost is introduced to simplify the scheduling model. Dominance conditions necessary for an optimal schedule are derived for this scheduling problem. Efficient methods exploiting these conditions are proposed for obtaining an optimal schedule minimizing the modified expected total cost. Numerical examples are demonstrated to show the effectiveness of the proposed methods.

An integrated plant loading model with economies of scale and scope

This paper presents a new formulation of a class of plant product mix-loading problems which are characterized by fixed facility costs, concave production costs and an integrated network structure which encompasses inbound supply and outbound distribution flows. In particular, we are interested in assigning product lines and volumes to a set of capacitated plants. Fixed costs are incurred when a product line is assigned to a plant. The production cost function also exhibits concavity with respect to each product line volume. Thus both scale and scope economies are considered explicitly in the model. The problem formulation leads to a concave mixed-integer mathematical program. We develop an optimization algorithm within the framework of Benders decomposition for the case of a piecewise linear concave cost function. Our algorithm generates optimal solutions efficiently. The problem solutions also illustrate how our model is effective in evaluating tradeoffs between inbound, production and outbound costs. Finally the model is used to illustrate the impact of various cost factors (logistics, scale and complexity) on optimal product mix solutions.

On properties of linear programming model for acid rain abatement

For linear programming problems arising in the field of acid rain abatement (in the case of a linear source-receptor relationship and linear or piecewise-linear abatement cost functions) the following properties are established. In the optimal solution, the number of sources which are not at an ‘inflection point’ of the corresponding piecewise-linear cost function is equal to the number of binding environmental constraints. (Here ‘inflection point’ denotes a point where the piecewise-linear cost function changes its slope; an environmental constrainst is called ‘binding’ if, for the optimal solution, the calculated deposition is just equal to, and not less than, the constraint.) As a consequences, several (i.e. the remaining) sources have optimum emissions which ‘get stuck’ at an inflection point. This influences the results of sensitivity analyses as well as the numerical accuracy of the optimal solution (if the piecewise-linear cost function has been obtained as a coarse approximation to a more complicated cost function). Moreover, for targets becoming more stringent, the optimum abatement costs may decrease for certain sources, although the total abatement costs increase, of course. This rather unexpected effect indicates that ‘pure’ optimization results should be applied, e.g. in negotiations, in a careful manner.

A Profit-maximizing Plant-loading Model with Demand Fill-rate Constraints

This paper discusses a new formulation of a class of plant product-mix loading problems which are characterized by capacitated production facilities, demand fill-rate requirements, fixed facility costs, concave variable production costs and an integrated network structure which encompasses inbound supply and outbound distribution flows. In particular, we are interested in assigning product lines and volumes to a set of capacitated plants under the demand fill-rate constraints. Fixed costs are incurred when a product line is assigned to a plant. The variable production-cost function also exhibits concavity with respect to each product-line volume. Thus both scale economies and plant focus effect are considered explicitly in the model. The model also can be used to determine which market to serve in order to best allocate the firm's resources. The problem formulation leads to a concave mixed-integer mathematical programme. Given the state of the art of non-linear programming techniques, it is often not possible to find global optima for reasonably sized problems. We develop an optimization algorithm within the framework of Benders' decomposition for the case of a piecewise linear concave cost function. Our algorithm generates optimal solutions efficiently.

The Optimal Control of Partially Observable Markov Processes over the Infinite Horizon: Discounted Costs

This paper treats the discounted cost, optimal control problem for Markov processes with incomplete state information. The optimization approach for these partially observable Markov processes is a generalization of the well-known policy iteration technique for finding optimal stationary policies for completely observable Markov processes. The state space for the problem is the space of state occupancy probability distributions (the unit simplex). The development of the algorithm introduces several new ideas, including the class of finitely transient policies, which are shown to possess piecewise linear cost functions. The paper develops easily implemented approximations to stationary policies based on these finitely transient policies and shows that the concave hull of an approximation can be included in the well-known Howard policy improvement algorithm with subsequent convergence. The paper closes with a detailed example illustrating the application of the algorithm to the two-state partially observable Markov process.

An Optimal Policy for Operating a Multipurpose Reservoir

This paper presents an analytically determined general optimal policy for operating a multipurpose reservoir and depicts it in graphical form. Piecewise linear cost functions are used to describe downstream flow costs, reservoir level costs, and costs incurred in meeting and not meeting power demands, the cost parameters involved in these functions being left arbitrary to allow wide applicability. The paper assumes random inflows into the reservoir, known power demands, and the availability of a finite-capacity auxiliary power source. It uses dynamic programming as an analytical, as opposed to numerical, tool in determining the general form of an optimal operating policy for this model.