418 research outputs found
Risk Minimization, Regret Minimization and Progressive Hedging Algorithms
This paper begins with a study on the dual representations of risk and regret
measures and their impact on modeling multistage decision making under
uncertainty. A relationship between risk envelopes and regret envelopes is
established by using the Lagrangian duality theory. Such a relationship opens a
door to a decomposition scheme, called progressive hedging, for solving
multistage risk minimization and regret minimization problems. In particular,
the classical progressive hedging algorithm is modified in order to handle a
new class of linkage constraints that arises from reformulations and other
applications of risk and regret minimization problems. Numerical results are
provided to show the efficiency of the progressive hedging algorithms.Comment: 21 pages, 2 figure
Convergence properties of nonmonotone spectral projected gradient methods
AbstractIn a recent paper, a nonmonotone spectral projected gradient (SPG) method was introduced by Birgin et al. for the minimization of differentiable functions on closed convex sets and extensive presented results showed that this method was very efficient. In this paper, we give a more comprehensive theoretical analysis of the SPG method. In doing so, we remove various boundedness conditions that are assumed in existing results, such as boundedness from below of f, boundedness of xk or existence of accumulation point of {xk}. If ∇f(·) is uniformly continuous, we establish the convergence theory of this method and prove that the SPG method forces the sequence of projected gradients to zero. Moreover, we show under appropriate conditions that the SPG method has some encouraging convergence properties, such as the global convergence of the sequence of iterates generated by this method and the finite termination, etc. Therefore, these results show that the SPG method is attractive in theory
Variable Metric Method for Unconstrained Multiobjective Optimization Problems
In this paper, we propose a variable metric method for unconstrained
multiobjective optimization problems (MOPs). First, a sequence of points is
generated using different positive definite matrices in the generic framework.
It is proved that accumulation points of the sequence are Pareto critical
points. Then, without convexity assumption, strong convergence is established
for the proposed method. Moreover, we use a common matrix to approximate the
Hessian matrices of all objective functions, along which, a new nonmonotone
line search technique is proposed to achieve a local superlinear convergence
rate. Finally, several numerical results demonstrate the effectiveness of the
proposed method
Barzilai-Borwein Descent Methods for Multiobjective Optimization Problems with Variable Trade-off Metrics
The imbalances and conditioning of the objective functions influence the
performance of first-order methods for multiobjective optimization problems
(MOPs). The latter is related to the metric selected in the direction-finding
subproblems. Unlike single-objective optimization problems, capturing the
curvature of all objective functions with a single Hessian matrix is
impossible. On the other hand, second-order methods for MOPs use different
metrics for objectives in direction-finding subproblems, leading to a high
per-iteration cost. To balance per-iteration cost and better curvature
exploration, we propose a Barzilai-Borwein descent method with variable metrics
(BBDMO\_VM). In the direction-finding subproblems, we employ a variable metric
to explore the curvature of all objectives. Subsequently, Barzilai-Borwein's
method relative to the variable metric is applied to tune objectives, which
mitigates the effect of imbalances. We investigate the convergence behaviour of
the BBDMO\_VM, confirming fast linear convergence for well-conditioned problems
relative to the variable metric. In particular, we establish linear convergence
for problems that involve some linear objectives. These convergence results
emphasize the importance of metric selection, motivating us to approximate the
trade-off of Hessian matrices to better capture the geometry of the problem.
Comparative numerical results confirm the efficiency of the proposed method,
even when applied to large-scale and ill-conditioned problems
The FCNC top-squark decay as a probe of squark mixing
In supersymmetry (SUSY) the flavor mixing between top-squark (stop) and
charm-squark (scharm) induces the flavor-changing neutral-current (FCNC) stop
decay . Searching for this decay serves as a
probe of soft SUSY breaking parameters. Focusing on the stop pair production
followed by the FCNC decay of one stop and the charge-current decay of the
other stop, we investigate the potential of detecting this FCNC stop decay at
the Fermilab Tevatron, the CERN Large Hadron Collider (LHC) and the
next-generation linear collider (LC). We find that this decay may not
be accessible at the Tevatron, but could be observable at the LHC and the LC
with high sensitivity.Comment: 13 pages, 3 figures (version to appear in PRD
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