60,161 research outputs found
Algebraic solutions of tropical optimization problems
We consider multidimensional optimization problems, which are formulated and
solved in terms of tropical mathematics. The problems are to minimize
(maximize) a linear or nonlinear function defined on vectors of a
finite-dimensional semimodule over an idempotent semifield, and may have
constraints in the form of linear equations and inequalities. The aim of the
paper is twofold: first to give a broad overview of known tropical optimization
problems and solution methods, including recent results; and second, to derive
a direct, complete solution to a new constrained optimization problem as an
illustration of the algebraic approach recently proposed to solve tropical
optimization problems with nonlinear objective function.Comment: 25 pages, presented at Intern. Conf. "Algebra and Mathematical Logic:
Theory and Applications", June 2-6, 2014, Kazan, Russi
Limits to the scope of applicability of extended formulations for LP models of combinatorial optimization problems: A summary
We show that new definitions of the notion of "projection" on which some of
the recent "extended formulations" works (such as Kaibel (2011); Fiorini et al.
(2011; 2012); Kaibel and Walter (2013); Kaibel and Weltge (2013) for example)
have been based can cause those works to over-reach in their conclusions in
relating polytopes to one another when the sets of the descriptive variables
for those polytopes are disjoint.Comment: 9 pages. arXiv admin note: substantial text overlap with
arXiv:1309.1823; This version (.v2) is simply a more formalized presentatio
Using tropical optimization to solve constrained minimax single-facility location problems with rectilinear distance
The aim of this paper is twofold: first, to extend the area of applications
of tropical optimization by solving new constrained location problems, and
second, to offer new closed-form solutions to general problems that are of
interest to location analysis. We consider a constrained minimax
single-facility location problem with addends on the plane with rectilinear
distance. The solution commences with the representation of the problem in a
standard form, and then in terms of tropical mathematics, as a constrained
optimization problem. We use a transformation technique, which can act as a
template to handle optimization problems in other application areas, and hence
is of independent interest. To solve the constrained optimization problem, we
apply methods and results of tropical optimization, which provide direct,
explicit solutions. The results obtained serve to derive new solutions of the
location problem, and of its special cases with reduced sets of constraints, in
a closed form, ready for practical implementation and immediate computation. As
illustrations, numerical solutions of example problems and their graphical
representation are given. We conclude with an application of the results to
optimal location of the central monitoring facility in an indoor video
surveillance system in a multi-floor building environment.Comment: 29 pages, 3 figure
Trading Regret for Efficiency: Online Convex Optimization with Long Term Constraints
In this paper we propose a framework for solving constrained online convex
optimization problem. Our motivation stems from the observation that most
algorithms proposed for online convex optimization require a projection onto
the convex set from which the decisions are made. While for
simple shapes (e.g. Euclidean ball) the projection is straightforward, for
arbitrary complex sets this is the main computational challenge and may be
inefficient in practice. In this paper, we consider an alternative online
convex optimization problem. Instead of requiring decisions belong to
for all rounds, we only require that the constraints which define
the set be satisfied in the long run. We show that our framework
can be utilized to solve a relaxed version of online learning with side
constraints addressed in \cite{DBLP:conf/colt/MannorT06} and
\cite{DBLP:conf/aaai/KvetonYTM08}. By turning the problem into an online
convex-concave optimization problem, we propose an efficient algorithm which
achieves regret bound and
bound for the violation of constraints. Then we
modify the algorithm in order to guarantee that the constraints are satisfied
in the long run. This gain is achieved at the price of getting
regret bound. Our second algorithm is based on
the Mirror Prox method \citep{nemirovski-2005-prox} to solve variational
inequalities which achieves bound for
both regret and the violation of constraints when the domain \K can be
described by a finite number of linear constraints. Finally, we extend the
result to the setting where we only have partial access to the convex set
and propose a multipoint bandit feedback algorithm with the same
bounds in expectation as our first algorithm
Convergence analysis of approximate primal solutions in dual first-order methods
Dual first-order methods are powerful techniques for large-scale convex
optimization. Although an extensive research effort has been devoted to
studying their convergence properties, explicit convergence rates for the
primal iterates have only been established under global Lipschitz continuity of
the dual gradient. This is a rather restrictive assumption that does not hold
for several important classes of problems. In this paper, we demonstrate that
primal convergence rate guarantees can also be obtained when the dual gradient
is only locally Lipschitz. The class of problems that we analyze admits general
convex constraints including nonlinear inequality, linear equality, and set
constraints. As an approximate primal solution, we take the minimizer of the
Lagrangian, computed when evaluating the dual gradient. We derive error bounds
for this approximate primal solution in terms of the errors of the dual
variables, and establish convergence rates of the dual variables when the dual
problem is solved using a projected gradient or fast gradient method. By
combining these results, we show that the suboptimality and infeasibility of
the approximate primal solution at iteration are no worse than
when the dual problem is solved using a projected gradient
method, and when a fast dual gradient method is used
Extremal properties of tropical eigenvalues and solutions to tropical optimization problems
An unconstrained optimization problem is formulated in terms of tropical
mathematics to minimize a functional that is defined on a vector set by a
matrix and calculated through multiplicative conjugate transposition. For some
particular cases, the minimum in the problem is known to be equal to the
tropical spectral radius of the matrix. We examine the problem in the common
setting of a general idempotent semifield. A complete direct solution in a
compact vector form is obtained to this problem under fairly general
conditions. The result is extended to solve new tropical optimization problems
with more general objective functions and inequality constraints. Applications
to real-world problems that arise in project scheduling are presented. To
illustrate the results obtained, numerical examples are also provided.Comment: 22 pages, presented at ILAS 2013 Conference (Providence, RI, 2013),
major revisio
A constrained tropical optimization problem: complete solution and application example
The paper focuses on a multidimensional optimization problem, which is
formulated in terms of tropical mathematics and consists in minimizing a
nonlinear objective function subject to linear inequality constraints. To solve
the problem, we follow an approach based on the introduction of an additional
unknown variable to reduce the problem to solving linear inequalities, where
the variable plays the role of a parameter. A necessary and sufficient
condition for the inequalities to hold is used to evaluate the parameter,
whereas the general solution of the inequalities is taken as a solution of the
original problem. Under fairly general assumptions, a complete direct solution
to the problem is obtained in a compact vector form. The result is applied to
solve a problem in project scheduling when an optimal schedule is given by
minimizing the flow time of activities in a project under various activity
precedence constraints. As an illustration, a numerical example of optimal
scheduling is also presented.Comment: 20 pages, accepted for publication in Contemporary Mathematic
Optimal Covariance Control for Stochastic Systems Under Chance Constraints
This work addresses the optimal covariance control problem for stochastic
discrete-time linear time-varying systems subject to chance constraints.
Covariance steering is a stochastic control problem to steer the system state
Gaussian distribution to another Gaussian distribution while minimizing a cost
function. To the best of our knowledge, covariance steering problems have never
been discussed with probabilistic chance constraints although it is a natural
extension. In this work, first we show that, unlike the case with no chance
constraints, the covariance steering with chance constraints problem cannot
decouple the mean and covariance steering sub-problems. Then we propose an
approach to solve the covariance steering with chance constraints problem by
converting it to a semidefinite programming problem. The proposed algorithm is
verified using two simple numerical simulations
Bayesian optimization under mixed constraints with a slack-variable augmented Lagrangian
An augmented Lagrangian (AL) can convert a constrained optimization problem
into a sequence of simpler (e.g., unconstrained) problems, which are then
usually solved with local solvers. Recently, surrogate-based Bayesian
optimization (BO) sub-solvers have been successfully deployed in the AL
framework for a more global search in the presence of inequality constraints;
however, a drawback was that expected improvement (EI) evaluations relied on
Monte Carlo. Here we introduce an alternative slack variable AL, and show that
in this formulation the EI may be evaluated with library routines. The slack
variables furthermore facilitate equality as well as inequality constraints,
and mixtures thereof. We show how our new slack "ALBO" compares favorably to
the original. Its superiority over conventional alternatives is reinforced on
several mixed constraint examples.Comment: 24 pages, 5 figure
Combining Convex-Concave Decompositions and Linearization Approaches for solving BMIs, with application to Static Output Feedback
A novel optimization method is proposed to minimize a convex function subject
to bilinear matrix inequality (BMI) constraints. The key idea is to decompose
the bilinear mapping as a difference between two positive semidefinite convex
mappings. At each iteration of the algorithm the concave part is linearized,
leading to a convex subproblem.Applications to various output feedback
controller synthesis problems are presented. In these applications the
subproblem in each iteration step can be turned into a convex optimization
problem with linear matrix inequality (LMI) constraints. The performance of the
algorithm has been benchmarked on the data from COMPleib library.Comment: 22 page
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