115 research outputs found
Advances in multi-parametric mixed-integer programming and its applications
At many stages of process engineering we are confronted with data that have not yet revealed
their true values. Uncertainty in the underlying mathematical model of real processes
is common and poses an additional challenge on its solution. Multi-parametric programming
is a powerful tool to account for the presence of uncertainty in mathematical models. It
provides a complete map of the optimal solution of the perturbed problem in the parameter
space.
Mixed integer linear programming has widespread application in process engineering such
as process design, planning and scheduling, and the control of hybrid systems. A particular difficulty arises, significantly increasing the complexity and computational effort in retrieving
the optimal solution of the problem, when uncertainty is simultaneously present in the coefficients of the objective function and the constraints, yielding a general multi-parametric
(mp)-MILP problem.
In this thesis, we present novel solution strategies for this class of problems. A global
optimization procedure for mp-MILP problems, which adapts techniques from the deterministic
case to the multi-parametric framework, has been developed. One of the challenges
in multi-parametric global optimization is that parametric profiles, and not scalar values
as in the deterministic case, need to be compared. To overcome the computational burden
to derive a globally optimal solution, two-stage methods for the approximate solution of
mp-MILP problems are proposed. The first approach combines robust optimization and
multi-parametric programming; whereas in the second approach suitable relaxations of bilinear
terms are employed to linearize the constraints during the approximation stage. The
choice of approximation techniques used in the two-stage method has impact on the conservatism
of the solution estimate that is generated. Lastly, multi-parametric programming
based two-stage methods are applied in pro-active short-term scheduling of batch processes
when faced with varied sources of uncertainty, such of price, demand and operational level
uncertainty.Open Acces
Multi-parametric Analysis for Mixed Integer Linear Programming: An Application to Transmission Planning and Congestion Control
Enhancing existing transmission lines is a useful tool to combat transmission
congestion and guarantee transmission security with increasing demand and
boosting the renewable energy source. This study concerns the selection of
lines whose capacity should be expanded and by how much from the perspective of
independent system operator (ISO) to minimize the system cost with the
consideration of transmission line constraints and electricity generation and
demand balance conditions, and incorporating ramp-up and startup ramp rates,
shutdown ramp rates, ramp-down rate limits and minimum up and minimum down
times. For that purpose, we develop the ISO unit commitment and economic
dispatch model and show it as a right-hand side uncertainty multiple parametric
analysis for the mixed integer linear programming (MILP) problem. We first
relax the binary variable to continuous variables and employ the Lagrange
method and Karush-Kuhn-Tucker conditions to obtain optimal solutions (optimal
decision variables and objective function) and critical regions associated with
active and inactive constraints. Further, we extend the traditional branch and
bound method for the large-scale MILP problem by determining the upper bound of
the problem at each node, then comparing the difference between the upper and
lower bounds and reaching the approximate optimal solution within the decision
makers' tolerated error range. In additional, the objective function's first
derivative on the parameters of each line is used to inform the selection of
lines to ease congestion and maximize social welfare. Finally, the amount of
capacity upgrade will be chosen by balancing the cost-reduction rate of the
objective function on parameters and the cost of the line upgrade. Our findings
are supported by numerical simulation and provide transmission line planners
with decision-making guidance
Bilevel Network Design
This chapter is devoted to network design problems involving conflicting agents, referred to as the designer and the users, respectively. Such problems are best cast into the framework of bilevel programming, where the designer anticipates the reaction or rational users to its course of action, and fits many situations of interest. In this chapter, we consider four applications of very different nature, with a special focus on algorithmic issues
A receding horizon event-driven control strategy for intelligent traffic management
AbstractIn this paper, the intelligent traffic management within a smart city environment is addressed by developing an ad-hoc model predictive control strategy based on an event-driven formulation. To this end, a constrained hybrid system description is considered for safety verification purposes and a low-demanding receding horizon controller is then derived by exploiting set-theoretic arguments. Simulations are performed on the train-gate benchmark system to show the effectiveness and benefits of the proposed methodology
Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case
The search problem of computing a Stackelberg (or leader-follower)equilibrium (also referred to as an optimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the optimistic and pessimistic versions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomial-time algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers' game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader's commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless . Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies—a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier—under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as a pessimistic bilevel programming problem, we show that, with two followers, the problem is NP-hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless . This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave quadratic function over an unbounded nonconvex feasible region defined by linear and quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-the-art methods, we propose an exact ad hoc algorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader's strategy where such value is attained, also an -approximate strategy where is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computational experiments on a well-established testbed of game instances
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