scholarly journals Linear semidefinite programming problems: regularisation and strong dual formulations

2020 ◽  
pp. 17-27
Author(s):  
Olga I. Kostyukova ◽  
Tatiana V. Tchemisova
Keyword(s):  
Semidefinite Programming ◽  
Original Problem ◽  
Equivalent Form ◽  
Strong Duality ◽  
Regularity Conditions ◽  
Index Set ◽  
Duality Property ◽  
Linear Problems ◽  
Implicit And Explicit ◽  
Strong Dual

Regularisation consists in reducing a given optimisation problem to an equivalent form where certain regularity conditions, which guarantee the strong duality, are fulfilled. In this paper, for linear problems of semidefinite programming (SDP), we propose a regularisation procedure which is based on the concept of an immobile index set and its properties. This procedure is described in the form of a finite algorithm which converts any linear semidefinite problem to a form that satisfies the Slater condition. Using the properties of the immobile indices and the described regularization procedure, we obtained new dual SDP problems in implicit and explicit forms. It is proven that for the constructed dual problems and the original problem the strong duality property holds true.

scholarly journals Online Linear Programming: Dual Convergence, New Algorithms, and Regret Bounds

2021 ◽  
Author(s):  
Xiaocheng Li ◽  
Yinyu Ye
Keyword(s):  
Linear Programming ◽  
Learning Algorithm ◽  
Equivalent Form ◽  
Sample Average Approximation ◽  
Smoothness Condition ◽  
Regularity Conditions ◽  
Linear Programs ◽  
Constraint Matrix ◽  
The Past ◽  
Dependent Learning

We study an online linear programming (OLP) problem under a random input model in which the columns of the constraint matrix along with the corresponding coefficients in the objective function are independently and identically drawn from an unknown distribution and revealed sequentially over time. Virtually all existing online algorithms were based on learning the dual optimal solutions/prices of the linear programs (LPs), and their analyses were focused on the aggregate objective value and solving the packing LP, where all coefficients in the constraint matrix and objective are nonnegative. However, two major open questions were as follows. (i) Does the set of LP optimal dual prices learned in the existing algorithms converge to those of the “offline” LP? (ii) Could the results be extended to general LP problems where the coefficients can be either positive or negative? We resolve these two questions by establishing convergence results for the dual prices under moderate regularity conditions for general LP problems. Specifically, we identify an equivalent form of the dual problem that relates the dual LP with a sample average approximation to a stochastic program. Furthermore, we propose a new type of OLP algorithm, action-history-dependent learning algorithm, which improves the previous algorithm performances by taking into account the past input data and the past decisions/actions. We derive an [Formula: see text] regret bound (under a locally strong convexity and smoothness condition) for the proposed algorithm, against the [Formula: see text] bound for typical dual-price learning algorithms, where n is the number of decision variables. Numerical experiments demonstrate the effectiveness of the proposed algorithm and the action-history-dependent design.

scholarly journals Strong Duality for Semidefinite Programming

1997 ◽  
Vol 7 (3) ◽  
pp. 641-662 ◽  
Author(s):  
Motakuri V. Ramana ◽  
Levent Tunçel ◽  
Henry Wolkowicz
Keyword(s):  
Semidefinite Programming ◽  
Strong Duality

scholarly journals CQ-free optimality conditions and strong dual formulations for a special conic optimization problem

2020 ◽  
Vol 8 (3) ◽  
pp. 668-683
Author(s):  
Olga Kostyukova ◽  
Tatiana V. Tchemisova
Keyword(s):  
Optimality Conditions ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Constraint Qualifications ◽  
Strong Duality ◽  
Conic Optimization ◽  
First Order ◽  
Primal And Dual ◽  
Linear Cost ◽  
Strong Dual

In this paper, we consider a special class of conic optimization problems, consisting of set-semidefinite(or K-semidefinite) programming problems, where the set K is a polyhedral convex cone. For these problems, we introduce the concept of immobile indices and study the properties of the set of normalized immobile indices and the feasible set. This study provides the main result of the paper, which is to formulate and prove the new first-order optimality conditions in the form of a criterion. The optimality conditions are explicit and do not use any constraint qualifications. For the case of a linear cost function, we reformulate the K-semidefinite problem in a regularized form and construct its dual. We show that the pair of the primal and dual regularized problems satisfies the strong duality relation which means that the duality gap is vanishing.

scholarly journals APPLICATION OF SEMIDEFINITE PROGRAMMING TO TRUSS DESIGN OPTIMIZATION / SANTVAROS OPTIMIZAVIMO UŽDAVINIŲ SPRENDIMAS TAIKANT PUSIAU APIBRĖŽTĄ PROGRAMAVIMĄ

2015 ◽  
Vol 7 (3) ◽  
pp. 280-284
Author(s):  
Rasa Giniūnaitė
Keyword(s):  
Semidefinite Programming ◽  
Optimization Problems ◽  
Positive Semidefinite ◽  
Algebraic Manipulation ◽  
Truss Structure ◽  
Convex Constraints ◽  
Positive Semidefinite Matrices ◽  
Combinatorial Optimization Problems ◽  
Linear Problems ◽  
The One

Semidefinite Programming (SDP) is a fairly recent way of solving optimization problems which are becoming more and more important in our fast moving world. It is a minimization of linear function over the intersection of the cone of positive semidefinite matrices with an affine space, i.e. non-linear but convex constraints. All linear problems and many engineering and combinatorial optimization problems can be expressed as SDP, so it is highly applicable. There are many packages that use different algorithms to solve SDP problems. They can be downloaded from internet and easily learnt how to use, two of these are SeDuMi and SDPT-3. In this paper truss structure optimization problem with the goal of minimizing the mass of the truss structure was solved. After doing some algebraic manipulation the problem was formulated suitably for Semidefinite Programming. SeDuMi and SDPT-3 packages were used to solve it. The choice of the initial solution had a great impact on the result using SeDuMi. The mass obtained using SDPT-3 was on average smaller than the one obtained using SeDuMi. Moreover, SDPT-3 worked more efficiently. However, the comparison of my approach and two versions of particle swarm optimization algorithm implied that semidefinite programming is in general more appropriate for solving such problems. Pusiau apibrėžtas programavimas yra iškiliojo optimizavimo posritis, kuriame tikslo funkcija tiesinė, o leistinoji sritis – pusiau teigiamai apibrėžtų matricų kūgio ir afininės erdvės sankirta. Tai gana naujas optimizavimo problemų sprendimo būdas, tačiau jau plačiai taikomas sprendžiant inžinerinius bei kombinatorinius optimizavimo uždavinius. Tokiems uždaviniams spręsti yra daug skirtingų paketų, taikančių įvairius algoritmus. Šiame darbe buvo naudojami SeDuMi ir SDPT-3 paketai, kuriuos, kaip ir daugumą kitų, galima parsisiųsti iš interneto. Tikslas buvo rasti minimalią santvaros masę atsižvelgiant į numatytus apribojimus. Naudojant SDPT-3 gauta optimali masė buvo vidutiniškai mažesnė nei naudojant SeDuMi. SDPT-3 veikė efektyviau ir pradinių sąlygų pasirinkimas neturėjo tokios didelės įtakos sprendiniui kaip naudojant SeDuMi paketą. Palyginus rezultatus su sprendiniais, gautais taikant dalelių spiečiaus optimizavimo algoritmą, nustatyta, kad tokio tipo uždaviniams pusiau apibrėžtas programavimas yra tinkamesnis.

Convergent and Predictive Validity of Implicit and Explicit Anxiety Measures as a Function of Specificity Similarity and Content Similarity

2008 ◽  
Vol 24 (4) ◽  
pp. 254-262 ◽  
Author(s):  
Tobias Gschwendner ◽  
Wilhelm Hofmann ◽  
Manfred Schmitt
Keyword(s):  
Implicit Association Test ◽  
Theoretical Models ◽  
Implicit Measures ◽  
Speech Anxiety ◽  
Behavioral Indicators ◽  
General Anxiety ◽  
Anxious Behavior ◽  
Content Similarity ◽  
Implicit And Explicit ◽  
Validation Strategy

In the present study we applied a validation strategy for implicit measures like the IAT, which complements multitrait-multimethod (MTMM) analyses. As the measurement method (implicit vs. explicit) and underlying representation format (associative vs. propositional) are often confounded, the validation of implicit measures has to go beyond MTMM analysis and requires substantive theoretical models. In the present study (N = 133), we employed such a model ( Hofmann, Gschwendner, Nosek, & Schmitt, 2005 ) and investigated two moderator constructs in the realm of anxiety: specificity similarity and content similarity. In the first session, different general and specific anxiety measures were administered, among them an Implicit Association Test (IAT) general anxiety, an IAT-spider anxiety, and an IAT that assesses speech anxiety. In the second session, participants had to deliver a speech and behavioral indicators of speech anxiety were measured. Results showed that (a) implicit and explicit anxiety measures correlated significantly only on the same specification level and if they measured the same content, and (b) specific anxiety measures best predicted concrete anxious behavior. These results are discussed regarding the validation of implicit measures.

Stories and the Self

2020 ◽  
Vol 32 (2) ◽  
pp. 47-58 ◽  
Author(s):  
Stefan Krause ◽  
Markus Appel
Keyword(s):  
Meta Analysis ◽  
The Self ◽  
The Other ◽  
Lower Degree ◽  
Contrast Effects ◽  
Self Perception ◽  
Self Perceptions ◽  
Other Hand ◽  
The One ◽  
Implicit And Explicit

Abstract. Two experiments examined the influence of stories on recipients’ self-perceptions. Extending prior theory and research, our focus was on assimilation effects (i.e., changes in self-perception in line with a protagonist’s traits) as well as on contrast effects (i.e., changes in self-perception in contrast to a protagonist’s traits). In Experiment 1 ( N = 113), implicit and explicit conscientiousness were assessed after participants read a story about either a diligent or a negligent student. Moderation analyses showed that highly transported participants and participants with lower counterarguing scores assimilate the depicted traits of a story protagonist, as indicated by explicit, self-reported conscientiousness ratings. Participants, who were more critical toward a story (i.e., higher counterarguing) and with a lower degree of transportation, showed contrast effects. In Experiment 2 ( N = 103), we manipulated transportation and counterarguing, but we could not identify an effect on participants’ self-ascribed level of conscientiousness. A mini meta-analysis across both experiments revealed significant positive overall associations between transportation and counterarguing on the one hand and story-consistent self-reported conscientiousness on the other hand.

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