31 
A New Interpolation Approach for Linearly Constrained Convex OptimizationEspinoza, Francisco 08 1900 (has links)
In this thesis we propose a new class of Linearly Constrained Convex Optimization
methods based on the use of a generalization of Shepard's interpolation formula. We
prove the properties of the surface such as the interpolation property at the boundary
of the feasible region and the convergence of the gradient to the null space of the
constraints at the boundary. We explore several descent techniques such as steepest
descent, two quasiNewton methods and the Newton's method. Moreover, we implement
in the Matlab language several versions of the method, particularly for the
case of Quadratic Programming with bounded variables. Finally, we carry out performance
tests against Matab Optimization Toolbox methods for convex optimization
and implementations of the standard logbarrier and activeset methods. We conclude
that the steepest descent technique seems to be the best choice so far for our
method and that it is competitive with other standard methods both in performance
and empirical growth order.

32 
A geometric approach to integer optimization and its application for reachability analysis in Petri nets. / CUHK electronic theses & dissertations collectionJanuary 2009 (has links)
Finding integer solutions to linear equations has various real world applications. In the thesis, we investigate its application to the reachability analysis of Petri nets. Introduced by Petri in 1962, Petri net has been a powerful mathematical formalism for modeling, analyzing and designing discrete event systems. In the research community of Petri nets, finding a feasible path from the initial state to the target state in Petri net, known as reachability analysis, is probably one of the most important and challenging subjects. The reachability algebraic analysis is equivalent to finding the nonnegative integer solutions to a fundamental equation constructed from the Petri net. We apply our algorithm in this thesis to reachability analysis of Petri net by finding the nonnegative integer solutions to the fundamental equation. / Finding the optimal binary solution to a quadratic object function is known as the Binary Quadratic Programming problem (BQP), which has been studied extensively in the last three decades. In this thesis, by investigating geometric features of the ellipse contour of a concave quadratic function, we derive new upper and lower bounding methods for BQP. Integrating these new bounding schemes into a proposed solution algorithm of a branchandbound type, we propose an exact solution method in solving general BQP with promising preliminary computational results. Meanwhile, by investigating some special structures of the second order matrix and linear term in BQP, several polynomial time algorithms are discussed to solve some special cases of BQP. / In the realm of integer programming, finding integer solutions to linear equations is another important research direction. The problem is proved to be NPComplete, and several algorithms have been proposed such as the algorithm based on linear Diophantine equations as well as the method based on Groebner bases. Unlike the traditional algorithms, the new efficient method we propose in this thesis is based on our results on zero duality gap and the cell enumeration of an arrangement of hyperplanes in discrete geometry. / Integer programming plays an important role in operations research and has a wide range of applications in various fields. There are a lot of research directions in the area of integer programming. In this thesis, two main topics will be investigated in details. One is to find the optimal binary solution to a quadratic object function, and the other is to find integer solutions to linear equations. / Gu, Shenshen. / Adviser: Wang Jun. / Source: Dissertation Abstracts International, Volume: 7301, Section: B, page: . / Thesis (Ph.D.)Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 98103). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

33 
On cardinality constrained optimization. / CUHK electronic theses & dissertations collectionJanuary 2009 (has links)
Although cardinality constraints naturally arise in many applications, e.g., in portfolio selection problems of choosing small number of assets from a large pool of stocks or dynamic portfolio selection problems with limited trading dates within a given time horizon and in subset selection of the regression analysis, the stateoftheart in cardinality constrained optimization has been stagnant up to this stage, largely due to the inherent combinatorial nature of such hard problems. We focus in this research on developing efficient and implementable solution algorithms for cardinality constrained optimization by investigating prominent structures and hidden properties of such problems. More specifically, we develop solution algorithms for four specific cardinality constrained optimization problems, including (i) the cardinality constrained linearquadratic control problem, (ii) the optimal control problem of linear switched system with limited number of switching, (iii) the time cardinality constrained dynamic mean variance portfolio selection problem, and (iv) cardinality constrained quadratic optimization problem. Taking advantages of a linearquadratic structure of cardinality constrained optimization problems, we strive for analytical solutions when possible. More specifically, we derive an analytical solution for problem (iii) and obtain for both problems (i) and (ii) semianalytical expressions of the solution governed by a family of Ricattilike equations, which still suffer an exponentially growing complexity. To achieve highperformance of the solution algorithm, we devise algorithms of a branch and bound (BnB) type with various tight and computationallycheap lower bounds achieved by identifying suitable SDP formulations and by exploiting geometric properties of the problem. We demonstrate efficiency of our proposed solution schemes evidenced from numerical experiments and present a firm stepforward in tackling this longstanding challenge of cardinality constrained optimization. / Gao, Jianjun. / Adviser: Duan Li. / Source: Dissertation Abstracts International, Volume: 7211, Section: B, page: . / Thesis (Ph.D.)Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 134142). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

34 
Cardinality constrained discretetime linearquadratic control.January 2005 (has links)
Gao Jianjun. / Thesis (M.Phil.)Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 7576). / Abstracts in English and Chinese. / Chapter 1  Introduction  p.1 / Chapter 2  Solution Framework Using Dynamic Programming  p.7 / Chapter 2.1  Difficulty of using dynamic programming  p.8 / Chapter 2.2  Scalarstate problems  p.12 / Chapter 2.3  Timeinvariant system  p.17 / Chapter 2.4  Illustrative example of a scalarstate problem  p.21 / Chapter 3  Cardinality Constrained Quadratic Optimization  p.26 / Chapter 3.1  Reformulation  p.27 / Chapter 3.2  NP hardness  p.31 / Chapter 3.3  Solving CCQP with an efficient branch and bound method  p.34 / Chapter 3.3.1  Efficient branch and bound algorithm  p.34 / Chapter 3.3.2  Geometrical interpretation of the proposed ranking order  p.48 / Chapter 3.3.3  Additional algorithmic ideas for enhancing computational efficiency  p.56 / Chapter 3.4  Numerical example and computational results  p.60 / Chapter 4  Summary and Future Work  p.73

35 
The mixedinteger bilinear programming problem with extensions to zeroone quadratic programsAdams, Warren Philip January 1985 (has links)
This research effort is concerned with a class of mathematical programming problems referred to as MixedInteger Bilinear Programming Problems. This class of problems, which arises in production, locationallocation, and distributionapplication contexts, may be considered as a discrete version of the wellknown Bilinear Programming Problem in that one set of decision variables is restricted to be binary valued. The structure of this problem is studied, and special cases wherein it is readily solvable are identified. For the more general case, a new linearization technique is introduced and demonstrated to lead to a tighter linear programming relaxation than obtained through available linearization methods. Based on this linearization, a composite Lagrangian relaxationimplicit enumerationcutting plane algorithm is developed. Extensive computational experience is provided to test the efficiency of various algorithmic strategies and the effects of problem data on the computational effort of the proposed algorithm.
The solution strategy developed for the MixedInteger Bilinear Programming Problem may be applied, with suitable modifications,. to other classes of mathematical programming problems: in particular, to the ZeroOne Quadratic Programming Problem. In what may be considered as an extension to the work performed on the MixedInteger Bilinear Programming Problem, a solution strategy based on an equivalent linear reformulation is developed for the ZeroOne Quadratic Programming Problem. The strategy is essentially an implicit enumeration algorithm which employs Lagrangian relaxation, Benders' cutting planes, and local explorations. Computational experience for this problem class is provided to justify the worth of the proposed linear reformulation and algorithm. / Ph. D.

36 
New solution approaches for the quadratic assignment problemFomeni, Franklin Djeumou 18 January 2012 (has links)
MSc., Faculty of Science, University of the Witwatersrand, 2011 / A vast array of important practical problems, in many di erent elds, can be modelled and
solved as quadratic assignment problems (QAP). This includes problems such as university
campus layout, forest management, assignment of runners in a relay team, parallel and
distributed computing, etc. The QAP is a di cult combinatorial optimization problem
and solving QAP instances of size greater than 22 within a reasonable amount of time
is still challenging. In this dissertation, we propose two new solution approaches to the
QAP, namely, a BranchandBound method and a discrete dynamic convexized method.
These two methods use the standard quadratic integer programming formulation of the
QAP. We also present a lower bounding technique for the QAP based on an equivalent
separable convex quadratic formulation of the QAP. We nally develop two di erent new
techniques for nding initial strictly feasible points for the interior point method used in
the BranchandBound method. Numerical results are presented showing the robustness
of both methods.

37 
Discrete Approximations, Relaxations, and Applications in Quadratically Constrained Quadratic ProgrammingBeach, Benjamin Josiah 02 May 2022 (has links)
We present works on theory and applications for Mixed Integer Quadratically Constrained Quadratic Programs (MIQCQP). We introduce new mixed integer programming (MIP)based relaxation and approximation schemes for general Quadratically Constrained Quadratic Programs (QCQP's), and also study practical applications of QCQP's and Mixedinteger QCQP's (MIQCQP).
We first address a challenging tank blending and scheduling problem regarding operations for a chemical plant. We model the problem as a discretetime nonconvex MIQCP, then approximate this model as a MILP using a discretizationbased approach. We combine a rolling horizon approach with the discretization of individual chemical property specifications to deal with long scheduling horizons, timevarying quality specifications, and multiple suppliers with discrete arrival times.
Next, we study optimization methods applied to minimizing forces for poses and movements of chained Stewart platforms (SPs). These SPs are parallel mechanisms that are stiffer, and more precise, on average, than their serial counterparts at the cost of a smaller range of motion. The robot will be used in concert with several other types robots to perform complex assembly missions in space. We develop algorithms and optimization models that can efficiently decide on favorable poses and movements that reduce force loads on the robot, hence reducing wear on this machine, and allowing for a larger workspace and a greater overall payload capacity.
In the third work, we present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions and formulate this approximation using mixedinteger programming (MIP). Combining this with a diagonal perturbation technique to convert a nonseparable quadratic function into a separable one, we present a mixedinteger convex quadratic relaxation for nonconvex quadratic optimization problems. We study the strength (or sharpness) of our formulation and the tightness of its approximation. We computationally demonstrate that our model outperforms existing MIP relaxations, and on hard instances can compete with stateoftheart solvers.
Finally, we study piecewise linear relaxations for solving quadratically constrained quadratic programs (QCQP's). We introduce new relaxation methods based on univariate reformulations of nonconvex variable products, leveraging the relaxation from the third work to model each univariate quadratic term. We also extend the NMDT approach (Castro, 2015) to leverage discretization for both variables in a bilinear term, squaring the resulting precision for the same number of binary variables. We then present various results related to the relative strength of the various formulations. / Doctor of Philosophy / First, we study a challenging longhorizon supply acquisition problem for a chemical plant. For this problem, constraints with products of variables are required to track raw material composition from supply carriers to storage tanks to the production feed. We apply a mixedinteger nonlinear program (MIP) approximation of the problem combined with a rolling planning scheme to obtain good solutions for industry problems within a reasonable time frame.
Next, we study optimization methods applied to a robot designed as a stack of Stewart platforms (SPs), which will be used in concert with several other types robots to perform complex space missions. When chaining these SPs together, we obtain a robot that is generally stiffer more precise than a classic robot arm, enabling their potential use for a variety of purposes. Our methods can efficiently decide on favorable poses and movements for the robot that reduce force loads on the robot, hence reducing wear on this machine, and allowing for a larger usable range of motion and a greater overall payload capacity.
Our final two works focus on MIPbased techniques for nonconvex QCQP's. In the first work, we break down the objective into an easytohandle term minus some squared terms. We then introduce an elegant new MIPbased approximation to handle these squared terms. We prove that this approximation has strong theoretical guarantees, then demonstrate that it is effective compared to other approximations.
In the second, we directly model each variable product term using a MIP relaxation. We introduce two new formulations to do this that build on previous formulations, increasing the accuracy with the same number of integer variables. We then prove a variety of useful properties about the presented formulations, then compare them computationally on two families of problems.

38 
Using linear programming to solve convex quadratic programming problemsIlyes, Amy Louise January 1993 (has links)
No description available.

39 
Sequential quadratic programmingbased contingency constrained optimal power flowPajic, Slobodan. January 2003 (has links)
Thesis (M.S.)Worcester Polytechnic Institute. / Keywords: contingency; interior point method; optimal power flow. Includes bibliographical references (p. 8283).

40 
Applications of Integer Quadratic Programming in Control and CommunicationAxehill, Daniel January 2005 (has links)
The main topic of this thesis is integer quadratic programming with applications to problems arising in the areas of automatic control and communication. One of the most widespread modern control principles is the discretetime method Model Predictive Control (MPC). The main advantage with MPC, compared to most other control principles, is that constraints on control signals and states can easily be handled. In each time step, MPC requires the solution of a Quadratic Programming (QP) problem. To be able to use MPC for large systems, and at high sampling rates, optimization routines tailored for MPC are used. In recent years, the range of application of MPC has been extended from constrained linear systems to socalled hybrid systems. Hybrid systems are systems where continuous dynamics interact with logic. When this extension is made, binary variables are introduced in the problem. As a consequence, the QP problem has to be replaced by a far more challenging Mixed Integer Quadratic Programming (MIQP) problem. Generally, for this type of optimization problems, the computational complexity is exponential in the number of binary optimization variables. In modern communication systems, multiple users share a socalled multiaccess channel, where the information sent by different users is separated by using almost orthogonal codes. Since the codes are not completely orthogonal, the decoded information at the receiver is slightly correlated between different users. Further, noise is added during the transmission. To estimate the information originally sent, a maximum likelihood problem involving binary variables is solved. The process of simultaneously estimating the information sent by multiple users is called multiuser detection. In this thesis, the problem to efficiently solve MIQP problems originating from MPC is addressed. Two different algorithms are presented. First, a polynomial complexity preprocessing algorithm for binary quadratic programming problems is presented. By using the algorithm, some, or all, binary variables can be computed efficiently already in the preprocessing phase. In simulations, the algorithm is applied to unconstrained MPC problems with a mixture of real and binary control signals. It has also been applied to the multiuser detection problem, where simulations have shown that the bit error rate can be significantly reduced by using the proposed algorithm as compared to using common suboptimal algorithms. Second, an MIQP algorithm tailored for MPC is presented. The algorithm uses a branch and bound method where the relaxed node problems are solved by a dual active set QP algorithm. In this QP algorithm, the KKTsystems are solved using Riccati recursions in order to decrease the computational complexity. Simulation results show that both the QP solver and the MIQP solver proposed have lower computational complexity than corresponding generic solvers. / Report code: LiUTEKLIC2005:71.

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