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Matrix Completion with Deterministic Sampling: Theories and Methods

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2021年第2期43卷 549-566页

作者： Liu, Guangcan Liu, Qingshan Yuan, Xiao-Tong Wang, Meng Nanjing Univ Informat Sci & Technol Sch Automat B DAT Nanjing 210044 Peoples R China Nanjing Univ Informat Sci & Technol Sch Automat CICAEET Nanjing 210044 Peoples R China Hefei Univ Technol Sch Comp Sci & Informat Engn 193 Tunxi Rd Hefei 230009 Anhui Peoples R China

In some significant applications such as data forecasting, the locations of missing entries cannot obey any non-degenerate distributions, questioning the validity of the prevalent assumption that the missing data is randomly chosen according to some probabilistic model. To break through the limits of random sampling, we explore in this paper the problem of real-valued matrix completion under the setup of deterministic sampling. We propose two conditions, isomeric condition and relative well-conditionedness, for guaranteeing an arbitrary matrix to be recoverable from a sampling of the matrix entries. It is provable that the proposed conditions are weaker than the assumption of uniform sampling and, most importantly, it is also provable that the isomeric condition is necessary for the completions of any partial matrices to be identifiable. Equipped with these new tools, we prove a collection of theorems for missing data recovery as well as convex/nonconvex matrix completion. Among other things, we study in detail a Schatten quasi-norm induced method termed isomeric dictionary pursuit (IsoDP), and we show that IsoDP exhibits some distinct behaviors absent in the traditional bilinear programs.

关键词： Coherence Tools Automation Information science Two dimensional displays Probabilistic logic Data models Matrix completion deterministic sampling identifiability isomeric condition relative well-conditionedness Schatten quasi-norm bilinear programming

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Enhanced intersection cutting-plane approach for linear complementarity problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1996年第1期90卷 183-201页

作者： Sherali, HD Krishnamurthy, RS AlKhayyal, FA GEORGIA INST TECHNOL SCH IND & SYST ENGN ATLANTA GA 30332 USA SIEMENS EMPROS MINNEAPOLIS MN USA

In this paper, we develop an enhanced intersection cutting-plane algorithm for solving a mixed integer 0-1 bilinear programming formulation of the linear complementarity problem (LCP). The matrix M associated with the LCP is not assumed to possess any special structure, except that the corresponding feasible region is assumed to be bounded. A procedure is described to generate cuts that are deeper versions of the Tuy intersection cuts, based on a relaxation of the usual polar set. The proposed algorithm then attempts to find an LCP solution in the process of generating either a single or a pair of such strengthened intersection cuts. The process of generating these cuts involves a vertex-ranking scheme that either finds an LCP solution, or else these cuts eliminate the entire feasible region leading to the conclusion that no LCP solution exists. Computational experience on various test problems is provided.

关键词： linear complementarity problems bilinear programming nonconvex programming cutting planes intersection cuts vertex ranking

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Algorithms for discrete and continuous multicommodity flow network interdiction problems

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IIE TRANSACTIONS 2007年第1期39卷 15-26页

作者： Lim, Churlzu Smith, J. Cole Univ Arizona Dept Syst & Ind Engn Tucson AZ 85721 USA

We consider a network interdiction problem on a multicommodity flow network, in which an attacker disables a set of network arcs in order to minimize the maximum profit that can be obtained from shipping commodities across the network. The attacker is assumed to have some budget for destroying (or "interdicting") arcs, and each arc is associated with a positive interdiction expense. In this paper, we examine problems in which interdiction must be discrete (i.e., each arc must either be left alone or completely destroyed), and in which interdiction can be continuous (the capacities of arcs may be partially reduced). For the discrete problem, we describe a linearized model for optimizing network interdiction that is similar to previous studies in the field, and compare it to a penalty model that does not require linearization constraints. For the continuous case, we prescribe an optimal partitioning algorithm along with a heuristic procedure for estimating the optimal objective function value. We demonstrate on a set of randomly generated test data that our penalty model for the discrete interdiction problem significantly reduces computational time when compared to that consumed by the linearization model.

关键词： network interdiction bilinear programming integer programming multicommodity flows

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JOINTLY CONSTRAINED BICONVEX programming

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MATHEMATICS OF OPERATIONS RESEARCH 1983年第2期8卷 273-286页

作者： ALKHAYYAL, FA FALK, JE GEORGE WASHINGTON UNIV DEPT OPERAT RESWASHINGTONDC 20052

This paper presents a branch-and-bound algorithm for minimizing the sum of a convex function in x, a convex function in y and a bilinear term in x and y over a closed set. Such an objective function is called biconvex with biconcave functions similarly defined. The feasible region of this model permits joint constraints in x and y to be expressed. The bilinear programming problem becomes a special case of the problem addressed in this paper. We prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point. The algorithm is proven to converge to a global solution of the nonconvex program. We discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs, for which the algorithm has been implemented.

关键词： biconvex functions. bilinear programming branch and bound algorithm global convergence nonconvex programming

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Linearized Robust Counterparts of Two-Stage Robust Optimization Problems with Applications in Operations Management

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INFORMS JOURNAL ON COMPUTING 2021年第3期33卷 1138-1161页

作者： Ardestani-Jaafari, Amir Delage, Erick Univ British Columbia Fac Management Kelowna BC V1V 1V7 Canada HEC Montreal Dept Decis Sci Montreal PQ H3T 2A7 Canada Grp Res Decis Anal GERAD Montreal PQ Canada

In this article, we discuss an alternative method for deriving conservative approximation models for two-stage robust optimization problems. The method mainly relies on a linearization scheme employed in bilinear programming;therefore, we will say that it gives rise to the linearized robust counterpart models. We identify a close relation between this linearized robust counterpart model and the popular affinely adjustable robust counterpart model. We also describe methods of modifying both types of models to make these approximations less conservative. These methods are heavily inspired by the use of valid linear and conic inequalities in the linearization process for bilinear models. We finally demonstrate how to employ this new scheme in location-transportation and multi-item newsvendor problems to improve the numerical efficiency and performance guarantees of robust optimization.

关键词： two-stage adjustable robust optimization affinely adjustable robust counterpart linear programming relaxation bilinear programming

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A LOCALIZATION AND REFORMULATION DISCRETE programming APPROACH FOR THE RECTILINEAR DISTANCE LOCATION-ALLOCATION PROBLEM

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DISCRETE APPLIED MATHEMATICS 1994年第1-3期49卷 357-378页

作者： SHERALI, HD RAMACHANDRAN, S KIM, SI KOREA UNIV DEPT IND ENGNSEOUL 136701SOUTH KOREA

This paper is concerned with the rectilinear distance location-allocation problem (RDLAP), which seeks the location of capacitated facilities, along with the allocation of their products to customers, so as to minimize total costs that are proportional to rectilinear distances and the amount shipped. Using a localization result, we present a new formulation of this problem as a mixed-integer bilinear programming problem. The problem is then reformulated using a reformulation linearization technique (RLT) as a linear mixed-integer problem that is shown to possess a tight linear programming relaxation. A branch-and-bound algorithm is then designed to implicitly enumerate over the location decision variable space. The special structure of the underlying linear programming relaxation is exploited to derive quick lower bounds via a suitable Lagrangian dual formulation. This methodology enables us to solve larger problem than heretofore solvable, and affords provably good quality heuristic solutions upon premature termination. An illustrative example and computational experience are provided to demonstrate the efficiency of the proposed algorithm.

关键词： RECTILINEAR DISTANCES LOCATION-ALLOCATION MIXED-INTEGER bilinear programming LOCALIZATION RESULTS REFORMULATION LINEARIZATION TECHNIQUE LAGRANGIAN DUALITY IMPLICIT ENUMERATION

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Conflict Resolution in Construction Projects Using Nonzero-Sum Fuzzy Bimatrix Games

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IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY-TRANSACTIONS OF CIVIL ENGINEERING 2018年第4期42卷 371-379页

作者： Sharif, Mehrdad Kerachian, Reza Univ Alberta Dept Civil & Environm Engn Edmonton AB Canada Univ Tehran Sch Civil Engn Tehran Iran Univ Tehran Coll Engn Ctr Excellence Engn & Management Civil Infrastruc Tehran Iran

The large damages that result from construction conflicts due to delays, cost overruns and decreasing productivity justify developing new models and decision support systems for resolving conflicts in large construction projects. Since most games take place in uncertain environments, the payoffs cannot be exactly assessed in many real-world problems. In these games, the uncertainty is mainly due to the inaccuracy of information and fuzzy comprehension of situations by players. In such cases, it is reasonable to model the problems as games with fuzzy payoffs. In this paper, two methodologies are proposed for conflict resolution in construction projects using nonzero-sum games with fuzzy payoffs. The first methodology transforms the original matrix game into a family of its -cut equivalents. The second methodology introduces fuzzy goals for payoffs in order to incorporate ambiguity of player's judgments. In this game, each player tries to maximize the degree of attainment of his fuzzy goal. The methodologies are applied to a large oil project in the Persian Gulf. The results show that the proposed methodology can be effectively used for resolving contractual conflicts between owners and contractors in a construction project.

关键词： Construction disputes Fuzzy bimatrix games Conflict resolution bilinear programming Fuzzy payoff

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A convex envelope formula for multilinear functions

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JOURNAL OF GLOBAL OPTIMIZATION 1997年第4期10卷 425-437页

作者： Rikun, AD E COAST PROD GRP HAMDENCT 06517

Convex envelopes of multilinear functions on a unit hypercube are polyhedral. This well-known fast makes the convex envelope approximation very useful in the linearization of non-linear 0-1 programming problems and in global bilinear optimization. This paper presents necessary and sufficient conditions for a convex envelope to be a polyhedral function and illustrates how these conditions may be used in constructing of convex envelopes. The main result of the paper is a simple analytical formula, which defines some faces of the convex envelope of a multilinear function. This formula proves to be a generalization of the well known convex envelope formula for multilinear monomial functions.

关键词： nonlinear 0-1 optimization linearization convex envelope concave extension bilinear programming global optimization

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Global maximization of a generalized concave multiplicative function

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2008年第1期137卷 105-120页

作者： Benson, H. P. Univ Florida Dept Informat & Decis Sci Gainesville FL 32611 USA

This article presents a branch-and-bound algorithm for globally solving the problem (P) of maximizing a generalized concave multiplicative function over a compact convex set. Since problem (P) does not seem to have been studied previously, the algorithm is apparently the first algorithm to be proposed for solving this problem. It works by globally solving a problem (P1) equivalent to problem (P). The branch-and-bound search undertaken by the algorithm uses rectangular partitioning and takes place in a space which typically has a much smaller dimension than the space to which the decision variables of problem (P) belong. Convergence of the algorithm is shown;computational considerations and benefits for users of the algorithm are given. A sample problem is also solved.

关键词： global optimization branch-and-bound algorithms multiplicative programming quadratic programming bilinear programming

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Observer-Based Output Feedback Control Using Invariant Polyhedral Sets for Fuzzy T-S Models Under Constraints

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JOURNAL OF CONTROL AUTOMATION AND ELECTRICAL SYSTEMS 2023年第4期34卷 752-765页

作者： Isidorio, Isaac D. Dorea, Carlos E. T. Castelan, Eugenio B. Univ Fed Rio Grande Do Norte UFRN CT DCA BR-59078900 Natal RN Brazil Univ Fed Santa Catarina DAS CTC UFSC Florianopolis SC Brazil

We propose a systematic design method to compute an observer-based output feedback controller for fuzzy Takagi-Sugeno (T-S) systems with unmeasurable premise variables, subject to state and control constraints, based on the theory of invariant sets. Sufficient conditions are established for a polyhedron defined in the augmented state-space (state + estimation error) to be positively invariant. From these conditions, a bilinear optimization problem is formulated for the simultaneous computation of the gains of the controller and of a positively invariant polyhedron guaranteeing the satisfaction of the constraints. Numerical examples illustrate the effectiveness of the method.

关键词： bilinear programming Constraints Fuzzy T-S models Invariant sets Observers Output feedback control

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