13,355 research outputs found
A Two-Level Approach to Large Mixed-Integer Programs with Application to Cogeneration in Energy-Efficient Buildings
We study a two-stage mixed-integer linear program (MILP) with more than 1
million binary variables in the second stage. We develop a two-level approach
by constructing a semi-coarse model (coarsened with respect to variables) and a
coarse model (coarsened with respect to both variables and constraints). We
coarsen binary variables by selecting a small number of pre-specified daily
on/off profiles. We aggregate constraints by partitioning them into groups and
summing over each group. With an appropriate choice of coarsened profiles, the
semi-coarse model is guaranteed to find a feasible solution of the original
problem and hence provides an upper bound on the optimal solution. We show that
solving a sequence of coarse models converges to the same upper bound with
proven finite steps. This is achieved by adding violated constraints to coarse
models until all constraints in the semi-coarse model are satisfied. We
demonstrate the effectiveness of our approach in cogeneration for buildings.
The coarsened models allow us to obtain good approximate solutions at a
fraction of the time required by solving the original problem. Extensive
numerical experiments show that the two-level approach scales to large problems
that are beyond the capacity of state-of-the-art commercial MILP solvers
Domain Decomposition for Stochastic Optimal Control
This work proposes a method for solving linear stochastic optimal control
(SOC) problems using sum of squares and semidefinite programming. Previous work
had used polynomial optimization to approximate the value function, requiring a
high polynomial degree to capture local phenomena. To improve the scalability
of the method to problems of interest, a domain decomposition scheme is
presented. By using local approximations, lower degree polynomials become
sufficient, and both local and global properties of the value function are
captured. The domain of the problem is split into a non-overlapping partition,
with added constraints ensuring continuity. The Alternating Direction
Method of Multipliers (ADMM) is used to optimize over each domain in parallel
and ensure convergence on the boundaries of the partitions. This results in
improved conditioning of the problem and allows for much larger and more
complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201
Using a conic bundle method to accelerate both phases of a quadratic convex reformulation
We present algorithm MIQCR-CB that is an advancement of method
MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving
mixed-integer quadratic programs and works in two phases: the first phase
determines an equivalent quadratic formulation with a convex objective function
by solving a semidefinite problem , and, in the second phase, the
equivalent formulation is solved by a standard solver. As the reformulation
relies on the solution of a large-scale semidefinite program, it is not
tractable by existing semidefinite solvers, already for medium sized problems.
To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm
within a Lagrangian duality framework for solving that substantially
speeds up the first phase. Moreover, this algorithm leads to a reformulated
problem of smaller size than the one obtained by the original MIQCR method
which results in a shorter time for solving the second phase.
We present extensive computational results to show the efficiency of our
algorithm
Partitioning Procedure for Polynomial Optimization: Application to Portfolio Decisions with Higher Order Moments
We consider the problem of finding the minimum of a real-valued multivariate polynomial function constrained in a compact set defined by polynomial inequalities and equalities. This problem, called polynomial optimization problem (POP), is generally nonconvex and has been of growing interest to many researchers in recent years. Our goal is to tackle POPs using decomposition. Towards this goal we introduce a partitioning procedure. The problem manipulations are in line with the pattern used in the Benders decomposition [1], namely relaxation preceded by projection. Stengle’s and Putinar’s Positivstellensatz are employed to derive the so-called feasibility and optimality constraints, respectively. We test the performance of the proposed method on a collection of benchmark problems and we present the numerical results. As an application, we consider the problem of selecting an investment portfolio optimizing the mean, variance, skewness and kurtosis of the portfolio.Polynomial optimization, Semidefinite relaxations, Positivstellensatz, Sum of squares, Benders decomposition, Portfolio optimization
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