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Optimization Day
Å «É¸×çbçÍ March 2, 2006
Sponsored by ÅðÜ2- (”–)
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Organized by
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Schedule of Programs
Place : M106, Mathematics Building
March 2 (Thursday) Speakers/Events 09:30 - 10:00 Registration
10:00 - 10:45 Jie Sun
10:45 - 11:00 BREAK
11:00 - 11:35 ŠV (Lai-Jiu Lin) 11:35 - 12:10 r:) (Ruey-Lin Sheu)
12:10 - 14:00 LUNCH
14:00 - 14:45 Paul Tseng 14:45 - 15:20 ŠC (Chih-Jen Lin)
15:20 - 15:40 BREAK
15:40 - 16:15 ˝Ç (Hang-Chin Lai) 16:15 - 16:50 Ð W (Hsing Paul Luh) 16:50 - 17:25 ØY (Hsien-Chung Wu)
17:45 DINNER
The price of anarchy under nonlinear and asymmetric costs
Jie Sun
Department of Decision Sciences National University of Singapore
Singapore [email protected]
Abstract. We derive new bounds for the price of anarchy under nonlinear and asym- metric costs. The bounds depend on an additional factor called the intrinsic cost of the system and therefore tend to be more accurate than the current bounds that are dependant only on the degree of asymmetry of the Jacobian and the degree of the nonlinearity of the cost function.
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Optimization Methods with Signal Denoising Applications
Paul Tseng
Department of Mathematics University of Washington
Seattle, WA 98195 U.S.A.
Abstract. We consider an approach to signal denoising whereby a linear combination of wavelet bases is fitted to the noisy signal by maximizing the likelihood function minus an l1-penalty on the coefficients of combination. The l1-penalty induces spar- sity of the coefficients and thus avoiding oversmoothing the fine scale features. For Gaussian noise, the maximization problem can be solved efficiently using a block coordinate minimization (BCM) method. For more general noise, the maximization problem can be solved using a primal-dual interior point method.
More generally, we consider the problem of minimizing the sum of a smooth func- tion and a block-separable convex function. We propose a method based on applying BCM to a quadratic approximation of the smooth function at each iteration. This method is cheaper to implement, and has stronger convergence properties than BCM.
In particular, it achieves linear convergence under a certain local error bound assump- tion. Numerical test results and comparison with MINOS 5.5.1 will be presented.
[The first part of the talk is joint work with Sylvain Sardy and Andrew Bruce.
The second part is joint work with Sangwoon Yun.]
Systems of Generalized Quasi-Variational Inclusion Problems With Applications to Variational Analysis
ŠV (Lai-Jiu Lin) Department of Mathematics
National Chang-Hua University of Education Chang-Hua 500, Taiwan
Abstract. In this talk, we study an existence theorem of systems of generalized quasi- variational inclusion problem. By this result, we establish the existence theorems of solutions of systems of generalized equations, systems of generalized quasiequilibrium problem, common fixed point theorem, systems of strong quasisaddle point, sys- tems of weak quasisaddle point, systems of minimax theorem, mathematical program with systems of variational inclusion constraints, mathematical program with equi- librium constraints, systems of bilevel problem, semi-infinite problem with systems of equilibrium constraints and mathematical program with mix-variational inequalities constraints.
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Projected Gradient Methods for Non-negative Matrix Factorization
ŠC (Chih-Jen Lin) Department of Computer Science
National Taiwan University Taipei 10617, Taiwan
Abstract. Non-negative matrix factorization (NMF) is a useful unsupervised learning method. It requires the solution of a non-convex bound-constrained optimization problem. This work proposes two projected-gradient methods for NMF. Compared to existing approaches, the new methods have sound optimization properties. We discuss their efficient implementations and show that one proposed method converges faster than the popular multiplicative update approach.
The Karush-Kuhn-Tucker Optimality Conditions for the Optimization Problem with Fuzzy-Valued Objective Function
ØY (Hsien-Chung Wu) Department of Mathematics National Kaohsiung Normal University
Kaohsiung 802, Taiwan [email protected]
Abstract. The KKT conditions for an optimization problem with fuzzy-valued ob- jective function will be derived in this talk. A solution concept of this optimization problem is proposed by considering an ordering relation on the class of all fuzzy num- bers. The proposed solution concept will follow from the similar solution concept, called nondominated solution, in the multiobjective programming problem. In order to consider the differentiation of an fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara differ- ence to define the difference of two fuzzy numbers. Under these settings, the KKT optimality conditions will be elicited naturally by introducing the Lagrange function multipliers.
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An application in comparison of queues by stochastic directional convexity
Ð W (Hsing Paul Luh) Department of Applied Mathematics
National Cheng Chi University Taipei 11605, Taiwan
Abstract. Second order properties of queues are important in design and analysis of service systems. In this talk we show that the blocking probability of M/M/C/N queue is increasing directionally convex in (λ, −µ), where λ is arrival rate and µ is service rate. Consider a heterogeneous queueing system with non-stationary arrival and service processes. The arrival and service rates alternate between two levels (λ1, µ1) and (λ2, µ2), spending an exponentially distributed amount of time with rate cαi in level i, i = 1, 2. When the system is in state i, the arrival rate is λi and the service rate is µi. Applying the increasing directional convexity result we show that the blocking probability is decreasing in c. In fact, we show that a stronger result holds: the number of blocked customers by time t is decreasing in c in stochastic convex ordering for any t ≥ 0. In particular, when N = ∞, the result is reduced to that the number of customers in system at any time t is decreasing in c in stochastic convex ordering.
Modified Dinkelbach-Type Algorithm for Generalized Fractional Programs with Infinitely Many Ratios
r:) (Ruey-Lin Sheu) Department of Mathematics National Cheng Kung University
Tainan 701, Taiwan [email protected]
Abstract. We extend the Dinkelbach-type algorithm of Crouzeix, Ferland, and Schaible to solve min-max fractional programs with infinitely many ratios. Parallel to the case with finitely many ratios, the task is to solve a sequence of continuous min-max problems,
P (αk) = min
x∈X( max
t∈T [ft(x) − αkgt(x)]),
until {αk} converges to the root of P (α) = 0. The solution of P (αk) is used to generate αk+1. However, calculating the exact optimal solution of P (αk) requires an extraordinary amount of work. To improve, we apply an entropic regularization method which allows us to solve each problem P (αk) incompletely, generating an approximate sequence {˜αk}, while retaining the linear convergence rate under mild assumptions. We present also numerical test results on the algorithm which indicate that the new algorithm is robust and promising.
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Optimality Conditions for Integral Functional Involving Control Problem
˝Ç (Hang-Chin Lai)
Department of Applied Mathematics Chung Yuan Christian University
Chung-Li 32023, Taiwan [email protected]
Abstract. The classical variational problem is to minimize the integral functional J : X −→ R as the form:
J(x) =
Z b
a g(t, x(t), ˙x(t))dt
subject to x ∈ X = C1(T, Rn), x(a) = α, x(b) = β in Rn, where the integrand g(t, x(t), ˙x(t)) : T × Rn × Rn −→ R is continuous in t ∈ T = [a, b] and having continuous partial derivative w.r.t. x and ˙x, x ∈ C1(T, Rn). It is known that the optimal solution x satisfies the Euler- Lagrange eq.
d
dtg˙x(x, ˙x)(t) − gx(x, ˙x)(t) = 0.
This result is extended to non-convex, non-locally Lipschitz integrand involving con- trol problem as the form:
(Pc) Min J(x, u) =
Z b
a g(t, x(t). ˙x(t), u(t)) dt such that ˙x ∈ F (t, x(t), u(t)), u(t) ∈ U(t) ⊂ Y which is reduced to an unconstrained problem:
(P ) Min J(x) = F (x, ˙x) =
Z b
a Lt(x, ˙x) dt
In this talk, we will find the optimality (necessary) condition which satisfies the generalized Euler- Lagrange Equation, that is, there exists an absolutely continuous function α ∈ A1X such that
( ˙α, α(t)) ∈ ∂oLt(z, ˙z)
for almost all t ∈ T where ∂oLt(z, ˙z) is Clarke subgradient differential provided that z is an optimal solution of (P ) and Lt(z, ˙z) is measurable in t satisfying the quasi locally Lipschitz at (z, ˙z). If Lt(·, ·) is continuous differentiable w.r.t. (s, v) ∈ X × X,
then d
dtLt, ˙x(z, ˙z) = L(t, x)(z, ˙z).
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