147 research outputs found
Lifted Polymatroid Inequalities for Mean-Risk Optimization with Indicator Variables
We investigate a mixed 0-1 conic quadratic optimization problem with
indicator variables arising in mean-risk optimization. The indicator variables
are often used to model non-convexities such as fixed charges or cardinality
constraints. Observing that the problem reduces to a submodular function
minimization for its binary restriction, we derive three classes of strong
convex valid inequalities by lifting the polymatroid inequalities on the binary
variables. Computational experiments demonstrate the effectiveness of the
inequalities in strengthening the convex relaxations and, thereby, improving
the solution times for mean-risk problems with fixed charges and cardinality
constraints significantly
Multi-Commodity Multi-Facility Network Design
We consider multi-commodity network design models, where capacity can be
added to the arcs of the network using multiples of facilities that may have
different capacities. This class of mixed-integer optimization models appears
frequently in telecommunication network capacity expansion problems, train
scheduling with multiple locomotive options, supply chain, and service network
design problems. Valid inequalities used as cutting planes in branch-and-bound
algorithms have been instrumental in solving large-scale instances. We review
the progress that has been done in polyhedral investigations in this area by
emphasizing three fundamental techniques. These are the metric inequalities for
projecting out continuous flow variables, mixed-integer rounding from
appropriate base relaxations and shrinking the network to a small -node
graph. The basic inequalities derived from arc-set, cut-set and partition
relaxations of the network are also extensively utilized with certain
modifications in robust and survivable network design problems
Network Design with Probabilistic Capacities
We consider a network design problem with random arc capacities and give a
formulation with a probabilistic capacity constraint on each cut of the
network. To handle the exponentially-many probabilistic constraints a
separation procedure that solves a nonlinear minimum cut problem is introduced.
For the case with independent arc capacities, we exploit the supermodularity of
the set function defining the constraints and generate cutting planes based on
the supermodular covering knapsack polytope. For the general correlated case,
we give a reformulation of the constraints that allows to uncover and utilize
the submodularity of a related function. The computational results indicate
that exploiting the underlying submodularity and supermodularity arising with
the probabilistic constraints provides significant advantages over the
classical approaches
Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra
We consider minimizing a conic quadratic objective over a polyhedron. Such
problems arise in parametric value-at-risk minimization, portfolio
optimization, and robust optimization with ellipsoidal objective uncertainty;
and they can be solved by polynomial interior point algorithms for conic
quadratic optimization. However, interior point algorithms are not well-suited
for branch-and-bound algorithms for the discrete counterparts of these problems
due to the lack of effective warm starts necessary for the efficient solution
of convex relaxations repeatedly at the nodes of the search tree.
In order to overcome this shortcoming, we reformulate the problem using the
perspective of the quadratic function. The perspective reformulation lends
itself to simple coordinate descent and bisection algorithms utilizing the
simplex method for quadratic programming, which makes the solution methods
amenable to warm starts and suitable for branch-and-bound algorithms. We test
the simplex-based quadratic programming algorithms to solve convex as well as
discrete instances and compare them with the state-of-the-art approaches. The
computational experiments indicate that the proposed algorithms scale much
better than interior point algorithms and return higher precision solutions. In
our experiments, for large convex instances, they provide up to 22x speed-up.
For smaller discrete instances, the speed-up is about 13x over a barrier-based
branch-and-bound algorithm and 6x over the LP-based branch-and-bound algorithm
with extended formulations
Submodularity in conic quadratic mixed 0-1 optimization
We describe strong convex valid inequalities for conic quadratic mixed 0-1
optimization. These inequalities can be utilized for solving numerous practical
nonlinear discrete optimization problems from value-at-risk minimization to
queueing system design, from robust interdiction to assortment optimization
through appropriate conic quadratic mixed 0-1 relaxations. The inequalities
exploit the submodularity of the binary restrictions and are based on the
polymatroid inequalities over binaries for the diagonal case. We prove that the
convex inequalities completely describe the convex hull of a single conic
quadratic constraint as well as the rotated cone constraint over binary
variables and unbounded continuous variables. We then generalize and strengthen
the inequalities by incorporating additional constraints of the optimization
problem. Computational experiments on mean-risk optimization with correlations,
assortment optimization, and robust conic quadratic optimization indicate that
the new inequalities strengthen the convex relaxations substantially and lead
to significant performance improvements
Path Cover and Path Pack Inequalities for the Capacitated Fixed-Charge Network Flow Problem
Capacitated fixed-charge network flows are used to model a variety of
problems in telecommunication, facility location, production planning and
supply chain management. In this paper, we investigate capacitated path
substructures and derive strong and easy-to-compute \emph{path cover and path
pack inequalities}. These inequalities are based on an explicit
characterization of the submodular inequalities through a fast computation of
parametric minimum cuts on a path, and they generalize the well-known flow
cover and flow pack inequalities for the single-node relaxations of
fixed-charge flow models. We provide necessary and sufficient facet conditions.
Computational results demonstrate the effectiveness of the inequalities when
used as cuts in a branch-and-cut algorithm
A Bound Strengthening Method for Optimal Transmission Switching in Power Systems
This paper studies the optimal transmission switching (OTS) problem for power
systems, where certain lines are fixed (uncontrollable) and the remaining ones
are controllable via on/off switches. The goal is to identify a topology of the
power grid that minimizes the cost of the system operation while satisfying the
physical and operational constraints. Most of the existing methods for the
problem are based on first converting the OTS into a mixed-integer linear
program (MILP) or mixed-integer quadratic program (MIQP), and then iteratively
solving a series of its convex relaxations. The performance of these methods
depends heavily on the strength of the MILP or MIQP formulations. In this
paper, it is shown that finding the strongest variable upper and lower bounds
to be used in an MILP or MIQP formulation of the OTS based on the big- or
McCormick inequalities is NP-hard. Furthermore, it is proven that unless P=NP,
there is no constant-factor approximation algorithm for constructing these
variable bounds. Despite the inherent difficulty of obtaining the strongest
bounds in general, a simple bound strengthening method is presented to
strengthen the convex relaxation of the problem when there exists a connected
spanning subnetwork of the system with fixed lines. The proposed method can be
treated as a preprocessing step that is independent of the solver to be later
used for numerical calculations and can be carried out offline before
initiating the solver. A remarkable speedup in the runtime of the mixed-integer
solvers is obtained using the proposed bound strengthening method for medium-
and large-scale real-world systems
A Conic Integer Programming Approach to Constrained Assortment Optimization under the Mixed Multinomial Logit Model
We consider the constrained assortment optimization problem under the mixed
multinomial logit model. Even moderately sized instances of this problem are
challenging to solve directly using standard mixed-integer linear optimization
formulations. This has motivated recent research exploring customized
optimization strategies and approximation techniques. In contrast, we develop a
novel conic quadratic mixed-integer formulation. This new formulation, together
with McCormick inequalities exploiting the capacity constraints, enables the
solution of large instances using commercial optimization software
A Scalable Semidefinite Relaxation Approach to Grid Scheduling
Determination of the most economic strategies for supply and transmission of
electricity is a daunting computational challenge. Due to theoretical barriers,
so-called NP-hardness, the amount of effort to optimize the schedule of
generating units and route of power, can grow exponentially with the number of
decision variables. Practical approaches to this problem involve legacy
approximations and ad-hoc heuristics that may undermine the efficiency and
reliability of power system operations, that are ever growing in scale and
complexity. Therefore, the development of powerful optimization methods for
detailed power system scheduling is critical to the realization of smart grids
and has received significant attention recently. In this paper, we propose for
the first time a computational method, which is capable of solving large-scale
power system scheduling problems with thousands of generating units, while
accurately incorporating the nonlinear equations that govern the flow of
electricity on the grid. The utilization of this accurate nonlinear model, as
opposed to its linear approximations, results in a more efficient and
transparent market design, as well as improvements in the reliability of power
system operations. We design a polynomial-time solvable third-order
semidefinite programming (TSDP) relaxation, with the aim of finding a near
globally optimal solution for the original NP-hard problem. The proposed method
is demonstrated on the largest available benchmark instances from real-world
European grid data, for which provably optimal or near-optimal solutions are
obtained
Accommodating new flights into an existing airline flight schedule
We present two novel approaches to alter a flight network for introducing new
flights while maximizing airline's profit. A key feature of the first approach
is to adjust the aircraft cruise speed to compensate for the block times of the
new flights, trading off flying time and fuel burn. In the second approach, we
introduce aircraft swapping as an additional mechanism to provide greater
flexibility in reducing the incremental fuel cost and adjusting the capacity.
The nonlinear fuel-burn function and the binary aircraft swap and assignment
decisions complicate the optimization problem significantly. We propose strong
mixed-integer conic quadratic formulations to overcome the computational
difficulties. The reformulations enable solving instances with 300 flights from
a major U.S. airline optimally within reasonable compute times
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