25,407 research outputs found
Relative Entropy Relaxations for Signomial Optimization
Signomial programs (SPs) are optimization problems specified in terms of
signomials, which are weighted sums of exponentials composed with linear
functionals of a decision variable. SPs are non-convex optimization problems in
general, and families of NP-hard problems can be reduced to SPs. In this paper
we describe a hierarchy of convex relaxations to obtain successively tighter
lower bounds of the optimal value of SPs. This sequence of lower bounds is
computed by solving increasingly larger-sized relative entropy optimization
problems, which are convex programs specified in terms of linear and relative
entropy functions. Our approach relies crucially on the observation that the
relative entropy function -- by virtue of its joint convexity with respect to
both arguments -- provides a convex parametrization of certain sets of globally
nonnegative signomials with efficiently computable nonnegativity certificates
via the arithmetic-geometric-mean inequality. By appealing to representation
theorems from real algebraic geometry, we show that our sequences of lower
bounds converge to the global optima for broad classes of SPs. Finally, we also
demonstrate the effectiveness of our methods via numerical experiments
A transformation method for constrained-function minimization
A direct method for constrained-function minimization is discussed. The method involves the construction of an appropriate function mapping all of one finite dimensional space onto the region defined by the constraints. Functions which produce such a transformation are constructed for a variety of constraint regions including, for example, those arising from linear and quadratic inequalities and equalities. In addition, the computational performance of this method is studied in the situation where the Davidon-Fletcher-Powell algorithm is used to solve the resulting unconstrained problem. Good performance is demonstrated for 19 test problems by achieving rapid convergence to a solution from several widely separated starting points
Convergence Rate of Frank-Wolfe for Non-Convex Objectives
We give a simple proof that the Frank-Wolfe algorithm obtains a stationary
point at a rate of on non-convex objectives with a Lipschitz
continuous gradient. Our analysis is affine invariant and is the first, to the
best of our knowledge, giving a similar rate to what was already proven for
projected gradient methods (though on slightly different measures of
stationarity).Comment: 6 page
State Transition Algorithm
In terms of the concepts of state and state transition, a new heuristic
random search algorithm named state transition algorithm is proposed. For
continuous function optimization problems, four special transformation
operators called rotation, translation, expansion and axesion are designed.
Adjusting measures of the transformations are mainly studied to keep the
balance of exploration and exploitation. Convergence analysis is also discussed
about the algorithm based on random search theory. In the meanwhile, to
strengthen the search ability in high dimensional space, communication strategy
is introduced into the basic algorithm and intermittent exchange is presented
to prevent premature convergence. Finally, experiments are carried out for the
algorithms. With 10 common benchmark unconstrained continuous functions used to
test the performance, the results show that state transition algorithms are
promising algorithms due to their good global search capability and convergence
property when compared with some popular algorithms.Comment: 18 pages, 28 figure
On global minimizers of quadratic functions with cubic regularization
In this paper, we analyze some theoretical properties of the problem of
minimizing a quadratic function with a cubic regularization term, arising in
many methods for unconstrained and constrained optimization that have been
proposed in the last years. First we show that, given any stationary point that
is not a global solution, it is possible to compute, in closed form, a new
point with a smaller objective function value. Then, we prove that a global
minimizer can be obtained by computing a finite number of stationary points.
Finally, we extend these results to the case where stationary conditions are
approximately satisfied, discussing some possible algorithmic applications.Comment: Optimization Letters (2018
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