9,483 research outputs found
A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems
This paper presents a new approach for the efficient solution of singular optimal control problems (SOCPs). A novel feature of the proposed method is that it does not require a priori knowledge of the structure of solution. At first, the SOCP is converted into a binary optimal control problem. Then, by utilising the pseudospectral method, the resulting problem is transcribed to a mixed-binary non-linear programming problem. This mixed-binary non-linear programming problem, which can be solved by well-known solvers, allows us to detect the structure of the optimal control and to compute the approximating solution. The main advantages of the present method are that: (1) without a priori information, the structure of optimal control is detected; (2) it produces good results even using a small number of collocation points; (3) the switching times can be captured accurately. These advantages are illustrated through a numerical implementation of the method on four examples. (c) 2017 Informa UK Limited, trading as Taylor & Francis Grou
A convex analysis approach to optimal controls with switching structure for partial differential equations
Optimal control problems involving hybrid binary-continuous control costs are
challenging due to their lack of convexity and weak lower semicontinuity.
Replacing such costs with their convex relaxation leads to a primal-dual
optimality system that allows an explicit pointwise characterization and whose
Moreau-Yosida regularization is amenable to a semismooth Newton method in
function space. This approach is especially suited for computing switching
controls for partial differential equations. In this case, the optimality gap
between the original functional and its relaxation can be estimated and shown
to be zero for controls with switching structure. Numerical examples illustrate
the effectiveness of this approach
A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives
We propose a direct numerical method for the solution of an optimal control
problem governed by a two-side space-fractional diffusion equation. The
presented method contains two main steps. In the first step, the space variable
is discretized by using the Jacobi-Gauss pseudospectral discretization and, in
this way, the original problem is transformed into a classical integer-order
optimal control problem. The main challenge, which we faced in this step, is to
derive the left and right fractional differentiation matrices. In this respect,
novel techniques for derivation of these matrices are presented. In the second
step, the Legendre-Gauss-Radau pseudospectral method is employed. With these
two steps, the original problem is converted into a convex quadratic
optimization problem, which can be solved efficiently by available methods. Our
approach can be easily implemented and extended to cover fractional optimal
control problems with state constraints. Five test examples are provided to
demonstrate the efficiency and validity of the presented method. The results
show that our method reaches the solutions with good accuracy and a low CPU
time.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Vibration and Control', available from
[http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised
03-Sept-2018; Accepted 12-Oct-201
A Scalable Algorithm For Sparse Portfolio Selection
The sparse portfolio selection problem is one of the most famous and
frequently-studied problems in the optimization and financial economics
literatures. In a universe of risky assets, the goal is to construct a
portfolio with maximal expected return and minimum variance, subject to an
upper bound on the number of positions, linear inequalities and minimum
investment constraints. Existing certifiably optimal approaches to this problem
do not converge within a practical amount of time at real world problem sizes
with more than 400 securities. In this paper, we propose a more scalable
approach. By imposing a ridge regularization term, we reformulate the problem
as a convex binary optimization problem, which is solvable via an efficient
outer-approximation procedure. We propose various techniques for improving the
performance of the procedure, including a heuristic which supplies high-quality
warm-starts, a preprocessing technique for decreasing the gap at the root node,
and an analytic technique for strengthening our cuts. We also study the
problem's Boolean relaxation, establish that it is second-order-cone
representable, and supply a sufficient condition for its tightness. In
numerical experiments, we establish that the outer-approximation procedure
gives rise to dramatic speedups for sparse portfolio selection problems.Comment: Submitted to INFORMS Journal on Computin
The Discrete Dantzig Selector: Estimating Sparse Linear Models via Mixed Integer Linear Optimization
We propose a novel high-dimensional linear regression estimator: the Discrete
Dantzig Selector, which minimizes the number of nonzero regression coefficients
subject to a budget on the maximal absolute correlation between the features
and residuals. Motivated by the significant advances in integer optimization
over the past 10-15 years, we present a Mixed Integer Linear Optimization
(MILO) approach to obtain certifiably optimal global solutions to this
nonconvex optimization problem. The current state of algorithmics in integer
optimization makes our proposal substantially more computationally attractive
than the least squares subset selection framework based on integer quadratic
optimization, recently proposed in [8] and the continuous nonconvex quadratic
optimization framework of [33]. We propose new discrete first-order methods,
which when paired with state-of-the-art MILO solvers, lead to good solutions
for the Discrete Dantzig Selector problem for a given computational budget. We
illustrate that our integrated approach provides globally optimal solutions in
significantly shorter computation times, when compared to off-the-shelf MILO
solvers. We demonstrate both theoretically and empirically that in a wide range
of regimes the statistical properties of the Discrete Dantzig Selector are
superior to those of popular -based approaches. We illustrate that
our approach can handle problem instances with p = 10,000 features with
certifiable optimality making it a highly scalable combinatorial variable
selection approach in sparse linear modeling
- …