Biconvex sets and optimization with biconvex functions: a survey and extensions

Gorski, Jochen; Pfeuffer, Frank; Klamroth, Kathrin

doi:10.1007/s00186-007-0161-1

Cited by 603 publications

(382 citation statements)

References 36 publications

Supporting

Mentioning

377

Contrasting

Order By: Relevance

“…According to Theorem 6.2.2 in [20], this implies that w * ≥ w for every feasible solution w to (13). By non-negativity of p 0 , w * must therefore maximize (13). Since (12) and (13) are equivalent, we have thus shown that w * maximizes (12).…”

Section: Theorem 32mentioning

confidence: 77%

“…Hence, the Banach fixed point theorem guarantees existence and uniqueness of w * ∈ R S . This vector w * is feasible in (13), and any feasible solution w ∈ R S to (13) satisfies w ≤ φ(π; w). According to Theorem 6.2.2 in [20], this implies that w * ≥ w for every feasible solution w to (13).…”

Section: Theorem 32mentioning

confidence: 99%

“…This vector w * is feasible in (13), and any feasible solution w ∈ R S to (13) satisfies w ≤ φ(π; w). According to Theorem 6.2.2 in [20], this implies that w * ≥ w for every feasible solution w to (13). By non-negativity of p 0 , w * must therefore maximize (13).…”

Section: Theorem 32mentioning

confidence: 99%

See 2 more Smart Citations

Robust Markov Decision Processes

2013

View full text Add to dashboard Cite

Markov decision processes (MDPs) are powerful tools for decision making in uncertain dynamic environments. However, the solutions of MDPs are of limited practical use due to their sensitivity to distributional model parameters, which are typically unknown and have to be estimated by the decision maker. To counter the detrimental effects of estimation errors, we consider robust MDPs that offer probabilistic guarantees in view of the unknown parameters. To this end, we assume that an observation history of the MDP is available. Based on this history, we derive a confidence region that contains the unknown parameters with a pre-specified probability 1 − β. Afterwards, we determine a policy that attains the highest worst-case performance over this confidence region. By construction, this policy achieves or exceeds its worst-case performance with a confidence of at least 1 − β. Our method involves the solution of tractable conic programs of moderate size.Keywords Robust Optimization; Markov Decision Processes; Semidefinite Programming.Notation For a finite set X = {1, . . . , X}, M(X ) denotes the probability simplex in RX . An X -valued random variable χ has distribution m ∈ M(X ), denoted by χ ∼ m, if P(χ = x) = m x for all x ∈ X . By default, all vectors are column vectors. We denote by e k the kth canonical basis vector, while e denotes the vector whose components are all ones. In both cases, the dimension will usually be clear from the context. For square matrices A and B, the relation A B indicates that the matrix A − B is positive semidefinite. We denote the space of symmetric n × n matrices by S n . The declaration f :implies that f is a continuous (affine) function from X to Y . For a matrix A, we denote its ith row by A i· (a row vector) and its jth column by A ·j .

show abstract

Section: Theorem 32mentioning

confidence: 77%

Section: Theorem 32mentioning

confidence: 99%

See 1 more Smart Citation

Robust Markov Decision Processes

2013

View full text Add to dashboard Cite

show abstract

“…For any threshold δ > 0, the algorithm terminates after finitely many iterations to a partial optimum of the chance constrained program (30), that is, a feasible point (x , χ , ρ ) where (x , ρ ) maximizes (30) for fixed χ and Q(χ , ρ ) represents the worst-case probability that the system is safe under the fixed decision x . For a convergence proof we refer to [32].…”

Section: Ambiguity Setmentioning

confidence: 99%

A distributionally robust perspective on uncertainty quantification and chance constrained programming

et al. 2015

View full text Add to dashboard Cite

The objective of uncertainty quantification is to certify that a given physical, engineering or economic system satisfies multiple safety conditions with high probability. A more ambitious goal is to actively influence the system so as to guarantee and maintain its safety, a scenario which can be modeled through a chance constrained program. In this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry, unimodality or independence patterns. We delineate the watershed between tractability and intractability in ambiguity-averse uncertainty quantification and chance constrained programming. Using tools from distributionally robust optimization, we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones.

show abstract

“…Note that one may recover the standard passive-aggressive algorithm by simply fixing the embedding matrix to the identity matrix. Also note that if the right stopping criteria are retained, Algorithm 1 is guaranteed to converge to a local optimum of (5) (see (Gorski et al, 2007)). …”

Section: An Alternating Online Proceduresmentioning

confidence: 99%

Online Learning of Task-specific Word Representations with a Joint Biconvex Passive-Aggressive Algorithm

Pelletier¹,

Ralaivola²

2017

Proceedings of the 15th Conference of the European Chapter of The Association for Computational Linguistics: Volume 1

View full text Add to dashboard Cite

This paper presents a new, efficient method for learning task-specific word vectors using a variant of the PassiveAggressive algorithm. Specifically, this algorithm learns a word embedding matrix in tandem with the classifier parameters in an online fashion, solving a biconvex constrained optimization at each iteration. We provide a theoretical analysis of this new algorithm in terms of regret bounds, and evaluate it on both synthetic data and NLP classification problems, including text classification and sentiment analysis. In the latter case, we compare various pre-trained word vectors to initialize our word embedding matrix, and show that the matrix learned by our algorithm vastly outperforms the initial matrix, with performance results comparable or above the state-of-the-art on these tasks.

show abstract

Biconvex sets and optimization with biconvex functions: a survey and extensions

Cited by 603 publications

References 36 publications

Robust Markov Decision Processes

Robust Markov Decision Processes

A distributionally robust perspective on uncertainty quantification and chance constrained programming

Online Learning of Task-specific Word Representations with a Joint Biconvex Passive-Aggressive Algorithm

Contact Info

Product

Resources

About