Mixed-integer Linear Constraints Research Articles

An accelerated extended cutting plane approach with piecewise linear approximations for signomial geometric programming

This paper presents a global optimization approach for solving signomial geometric programming (SGP) problems. We employ an accelerated extended cutting plane (ECP) approach integrated with piecewise linear (PWL) approximations to solve the global optimization of SGP problems. In this approach, we separate the feasible regions determined by the constraints into convex and nonconvex ones in the logarithmic domain. In the nonconvex feasible regions, the corresponding constraint functions are converted into mixed integer linear constraints using PWL approximations, while the other constraints with convex feasible regions are handled by the ECP method. We also use pre-processed initial cuts and batched cuts to accelerate the proposed algorithm. Numerical results show that the proposed approach can solve the global optimization of SGP problems efficiently and effectively.

A Characterization of the SSD-Efficient Frontier of Portfolio Weights by Means of a Set of Mixed-Integer Linear Constraints

In this paper, the set of all second-order stochastic dominance (SSD)-efficient portfolios is characterized by using a series of mixed-integer linear constraints. Our derivation employs a combination of the first-order conditions of the utility maximization problem together with a judicious use of binary variables. This result opens the door to the formulation of optimizations whose objective function is free to select a particular portfolio out of the entire SSD-efficient set. This paper was accepted by Jerome Detemple, finance.

Mixed Integer Linear Fractional Programming for Conjunctive Use of Surface Water and Groundwater

AbstractA conjunctive-use model is developed for management of groundwater and surface water resources via mixed integer linear fractional programming (MILFP). The objective of the conjunctive-use model is to maximize the ratio of groundwater usage to surface water usage through a water supply network. A conditional head constraint is imposed to the conjunctive-use model to maintain aquifer sustainability. A transformation approach is introduced to transform the conditional head constraint into a set of mixed integer linear constraints in terms of groundwater head. Groundwater head is further linearized with respect to pumping rates that are decision variables. Eventually, the conjunctive-use model is to solve a successive MILFP problem by updating the response matrix in each iteration. To make an MILFP problem tractable, the study develops a transformation technique along with the Charnes–Cooper transformation approach to transform an MILFP problem into an equivalent problem of mixed integer linear progr...

Data-driven construction of Convex Region Surrogate models

With the increasing trend of solving more complex and integrated optimization problems, there is a need for developing process models that are sufficiently accurate as well as computationally efficient. In this work, we develop an algorithm for the data-driven construction of a type of surrogate model that can be formulated as a set of mixed-integer linear constraints, yet still provide good approximations of nonlinearities and nonconvexities. In such a surrogate model, which we refer to as Convex Region Surrogate (CRS), the feasible region is given by the union of convex regions in the form of polytopes, and for each region, the corresponding cost function can be approximated by a linear function. The general problem is as follows: given a set of data points in the parameter space and a scalar cost value associated with each data point, find a CRS model that approximates the feasible region and cost function indicated by the given data points. We present a two-phase algorithm to solve this problem and demonstrate its effectiveness with an extensive computational study as well as a real-world case study.

Operating room scheduling with Generalized Disjunctive Programming

This paper addresses the short-term scheduling problem involved in the selection of a subset of elective surgeries from a large waiting list. In order to overcome the combinatorial complexity, a decomposition algorithm is proposed that relies on two continuous-time Generalized Disjunctive Programming (GDP) models. More specifically, there is an upper-level planning model to select surgical assignments to operating rooms and a lower-level constrained scheduling model to synchronize surgeons operating in different rooms on a given day. The GDP models are reformulated using standard convex hull and big-M techniques so as to generate the most efficient set of integer or mixed-integer linear programming constraints. Through the solution of a set of real-life instances from the literature, we show that the new algorithm outperforms a full-space discrete-time formulation and a genetic algorithm, improving the total surgical time as well as the number of performed surgeries by 5%.

Robust Model Predictive Control for Signal Temporal Logic Synthesis

Most automated systems operate in uncertain or adversarial conditions, and have to be capable of reliably reacting to changes in the environment. The focus of this paper is on automatically synthesizing reactive controllers for cyber-physical systems subject to signal temporal logic (STL) specifications. We build on recent work that encodes STL specifications as mixed integer linear constraints on the variables of a discrete-time model of the system and environment dynamics. To obtain a reactive controller, we present solutions to the worst-case model predictive control (MPC) problem using a suite of mixed integer linear programming techniques. We demonstrate the comparative effectiveness of several existing worst-case MPC techniques, when applied to the problem of control subject to temporal logic specifications; our empirical results emphasize the need to develop specialized solutions for this domain.

Hybrid Model Predictive Control for Sequential Decision Policies in Adaptive Behavioral Interventions.

Control engineering offers a systematic and efficient method to optimize the effectiveness of individually tailored treatment and prevention policies known as adaptive or "just-in-time" behavioral interventions. The nature of these interventions requires assigning dosages at categorical levels, which has been addressed in prior work using Mixed Logical Dynamical (MLD)-based hybrid model predictive control (HMPC) schemes. However, certain requirements of adaptive behavioral interventions that involve sequential decision making have not been comprehensively explored in the literature. This paper presents an extension of the traditional MLD framework for HMPC by representing the requirements of sequential decision policies as mixed-integer linear constraints. This is accomplished with user-specified dosage sequence tables, manipulation of one input at a time, and a switching time strategy for assigning dosages at time intervals less frequent than the measurement sampling interval. A model developed for a gestational weight gain (GWG) intervention is used to illustrate the generation of these sequential decision policies and their effectiveness for implementing adaptive behavioral interventions involving multiple components.

Mixed Integer Linear Programming for UAV Trajectory Planning Problem

This paper describes a method for vehicles flying Trajectory Planning Problem in 3D environments. These requirements lead to non-convex constraints and difficult optimizations. It is shown that this problem can be rewritten as a linear program with mixed integer linear constraints that account for the collision avoidance used in model predictive control, running in real-time to incorporate feedback and compensate for uncertainty. An example is worked out in a real-time scheme, solved on-line to compensate for the effect of uncertainty as the maneuver progresses. In particular, we compare receding horizon control with arrival time approaches.

A Mixed Integer Linear Programming model of a zonal electricity market with a dominant producer

We consider a liberalized electricity market, divided in zones interconnected by capacitated transmission links, where a large dimensional power producer operates. We introduce a model for determining the optimal bidding strategies of the large dimensional producer, so as to maximize his own market share, while guaranteeing an annual profit target and satisfying technical constraints. The model determines the optimal medium-term resource scheduling and yields the hourly zonal electricity prices, as it includes constraints representing the Market Clearing process. In order to compute the global solution, the complementarity conditions are formulated as mixed integer linear constraints and the revenue terms are expressed by piece-wise linear functions. The model can be used for analyzing the behavior of market prices in electricity markets where a large dimensional producer can exert market power. It can also be used by investors as a simulation tool for evaluating both the impact on the market and the profitability of investment decisions in the zonal electricity market. A case study related to the Italian electricity market is discussed.

Optimization of Discrete Event Systems using Extended Finite Automata and Mixed-Integer Nonlinear Programming

AbstractThis paper presents a concept for converting a discrete event model, modeled with Extended Finite Automata (EFA), to mixed-integer linear constraints. The conversion handles the structure of modular EFAs, synchronization of EFAs using shared events and EFA execution order due to logical transition conditions. The paper also presents methods to reduce the number of variables and constraints by automatically analyzing the EFA model and the resulting problem formulation. An example of this is the special case of transition conditions used to model mutual exclusion of shared resources, where the conversion results in a significantly reduced problem formulation. The objective function is then built by summarizing weighted state cost functions and the result is a Mixed-Integer Nonlinear Programming problem. The main contribution of this paper is hence the combination of the simplicity in modeling a system with EFAs and an efficient formulation of the optimization problem that can be solved by standard optimization software.

Non-Transferable Utility Coalitional Games via Mixed-Integer Linear Constraints

Coalitional games serve the purpose of modeling payoff distribution problems in scenarios where agents can collaborate by forming coalitions in order to obtain higher worths than by acting in isolation. In the classical Transferable Utility (TU) setting, coalition worths can be freely distributed amongst agents. However, in several application scenarios, this is not the case and the Non-Transferable Utility setting (NTU) must be considered, where additional application-oriented constraints are imposed on the possible worth distributions. In this paper, an approach to define NTU games is proposed which is based on describing allowed distributions via a set of mixed-integer linear constraints applied to an underlying TU game. It is shown that such games allow non-transferable conditions on worth distributions to be specified in a natural and succinct way. The properties and the relationships among the most prominent solution concepts for NTU games that hold when they are applied on (mixed-integer) constrained games are investigated. Finally, a thorough analysis is carried out to assess the impact of issuing constraints on the computational complexity of some of these solution concepts.

Non-transferable utility coalitional games via mixed-integer linear constraints

Coalitional games serve the purpose of modeling payoff distribution problems in scenarios where agents can collaborate by forming coalitions in order to obtain higher worths than by acting in isola...

Three-Dimensional Path Planning of a Climbing Robot Using Mixed Integer Linear Programming

The City-Climber robot is a novel wall-climbing robot developed at The City College of New York that has the capability to move on floors, climb walls, walk on ceilings and transit between them. In this paper, we first develop the dynamic model of the City-Climber robot when it travel on different surfaces, i.e., floors, walls and ceilings, respectively. Then, we present a path planning method for the City-Climber robot using mixed integer linear programming (MILP) in three-dimensional (3-D) building environments that consist of objects with primitive geometrical shapes. MILP provides an optimization framework that can directly incorporate dynamic constraints with logical constraints such as obstacle avoidance and waypoint selection. In order to use MILP to solve the obstacle avoidance problem, we simplify and decouple the robot dynamic model into a linear system by introducing a restricting admissible controller. The decoupled model and obstacle can be rewritten as a linear program with mixed-integer linear constraints that account for the collision avoidance. A key benefit of this approach is that the path optimization can be readily solved using the AMPL and CPLEX optimization software with a MATLAB interface. Simulation results show that the framework of MILP is well suited for path planning and obstacle avoidance problems for the wall-climbing robot in 3-D environments.

A Mathematical Logic Approach for the Transformation of the Linear Conditional Piecewise Functions of Dispersion-and-Store and Cell Transmission Traffic Flow Models into Linear Mixed-Integer Form

The modeling of traffic control systems for solving such problems as surface street signalization, dynamic traffic assignment, etc., typically results in the appearance of a conditional function. For example, the consistent representation of the outflow discharge at an approach of a signalized intersection implies a function that is conditional on the signal indication and the prevailing traffic conditions. Representing such functions by some sort of constraint(s), ideally linear, so as to be considered in the context of a mathematical programming problem, is a nontrivial task, most often resolved by adopting restrictive assumptions regarding real-life process behavior. To address this general problem, we develop two methodologies that are largely based on analogies from mathematical logic that provide a practical device for the transformation of a specific form of a linear conditional piecewise function into a mixed integer model (MIM), i.e., a set of mixed-integer linear inequality constraints. We show the applicability of these methodologies to transforming into a MIM virtually every possible conditional piecewise function that can be found when one is modeling transportation systems based on the widely adopted dispersion-and-store and cell transmission traffic flow models, as well as to analyzing existing MIMs for identifying and eliminating redundancies.

Dynamic Ramping in Unit Commitment

This paper provides the modeling of dynamic ramping in unit commitment. Ramping up/down limits could be a constant, stepwise, or piecewise linear function of generation dispatch. The dynamic ramping limit is modeled as mixed integer linear constraints, and unit commitment is solved by a mixed integer programming (MIP) solver. Test results show the impact of dynamic ramping on unit commitment and the importance of modeling dynamic ramping in the minimization of system costs while taking into consideration physical and financial constraints of power systems.

MATHEMATICAL LOGIC IN THE MODELING OF CONTROL SYSTEMS

The design of a control strategy involves the consideration of a mathematical model of the corresponding process and includes the solution of a control problem. In particular, optimal control problems are viewed and solved as mathematical programming problems by optimizing an objective function over a feasible region defined by constraints. Clearly, the process model must be expressed in constraint form. However, consistency assumptions regarding the process behavior lead to the consideration of conditional piece-wise functions or logical sentences in the corresponding model that are not in constraint form. For this, two methodologies based on analogies from mathematical logic for the transformation of the aforementioned functions into mixed-integer linear constraints have been developed, and they are presented by applying them in a flow function that is met in the modeling of surface street signal systems.

Modeling of Start-Up and Shut-Down Power Trajectories of Thermal Units

This paper presents a detailed formulation to model the power trajectories followed by a thermal unit during the start-up and shut-down processes, as well as the ramping limitations when increasing or decreasing power. This model is formulated using mixed-integer linear constraints, thereby overcoming some drawbacks of those approaches based on heuristics, dynamic programming or Lagrangian relaxation. The formulation presented can be used in different scheduling problems arising in centralized and deregulated frameworks. The self-scheduling problem faced by a thermal generator in a pool-based electric energy market is used to illustrate the proposed model. Finally, a realistic case study is analyzed to show the computational behavior of the proposed model.