The focus of this dissertation is the advancement of theory and computation related to exact precision mathematical programming. Optimization software based on floating-point arithmetic can return suboptimal or incorrect resulting because of round-off errors or the use of numerical tolerances. Exact or correct results are necessary for some applications. Implementing software entirely in rational arithmetic can be prohibitively slow. A viable alternative is the use of hybrid methods that use fast numerical computation to obtain approximate results that are then verified or corrected with safe or exact computation. We study fast methods for sparse exact rational linear algebra, which arises as a bottleneck when solving linear programming problems exactly. Output sensitive methods for exact linear algebra are studied. Finally, a new method for computing valid linear programming bounds is introduced and proven effective as a subroutine for solving mixed-integer linear programming problems exactly. Extensive computational results are presented for each topic.

Many supply chain problems involve discrete decisions in a dynamic environment. The inventory routing problem is an example that combines the dynamic control of inventory at various facilities in a supply chain with the discrete routing decisions of a fleet of vehicles that moves product between the facilities. We study these problems modeled as mixed-integer programs and propose a time decomposition based on approximate inventory valuation. We generate the approximate value function with an algorithm that combines data fitting, discrete optimization and dynamic programming methodology. Our framework allows the user to specify a class of piecewise linear, concave functions from which the algorithm chooses the value function. The use of piecewise linear concave functions is motivated by intuition, theory and practice. Intuitively, concavity reflects the notion that inventory is marginally more valuable the closer one is to a stock-out. Theoretically, piecewise linear concave functions have certain structural properties that also hold for finite mixed-integer program value functions. (Whether the same properties hold in the infinite case is an open question, to our knowledge.) Practically, piecewise linear concave functions are easily embedded in the objective function of a maximization mixed-integer or linear program, with only a few additional auxiliary continuous variables. We evaluate the solutions generated by our value functions in a case study using maritime inventory routing instances inspired by the petrochemical industry. The thesis also includes two other contributions. First, we review various data fitting optimization models related to piecewise linear concave functions, and introduce new mixed-integer programming formulations for some cases. The formulations may be of independent interest, with applications in engineering, mixed-integer non-linear programming, and other areas. Second, we study a discounted, infinite-horizon version of the canonical single-item lot-sizing problem and characterize its value function, proving that it inherits all properties of interest from its finite counterpart. We then compare its optimal policies to our algorithm's solutions as a proof of concept.

Multinomial selection problem procedures are ranking and selection techniques that aim to select the best (most probable) alternative based upon a sequence of multinomial observations. The classical formulation of the procedure design problem is to find a decision rule for terminating sampling. The decision rule should minimize the expected number of observations taken while achieving a specified indifference zone requirement on the prior probability of making a correct selection when the alternative configurations are in a particular subset of the probability space called the preference zone. We study the constrained version of the design problem in which there is a given maximum number of allowed observations. Numerous procedures have been proposed over the past 50 years, all of them suboptimal. In this thesis, we find via linear programming the optimal selection procedure for any given probability configuration. The optimal procedure turns out to be necessarily randomized in many cases. We also find via mixed integer programming the optimal non-randomized procedure. We demonstrate the performance of the methodology on a number of examples. We then reformulate the mathematical programs to make them more efficient to implement, thereby significantly expanding the range of computationally feasible problems. We prove that there exists an optimal policy which has at most one randomized decision point and we develop a procedure for finding such a policy. We also extend our formulation to replicate existing procedures. Next, we show that there is very little difference between the relative performances of the optimal randomized and non-randomized procedures. Additionally, we compare existing procedures using the optimal procedure as a benchmark, and produce updated tables for a number of those procedures. Then, we develop a methodology that guarantees the optimal randomized and non-randomized procedures for a broad class of variable observation cost functions -- the first of its kind. We examine procedure performance under a variety of cost functions, demonstrating that incorrect assumptions regarding marginal observation costs may lead to increased total costs. Finally, we investigate and challenge key assumptions concerning the indifference zone parameter and the conditional probability of correct selection, revealing some interesting implications.

<p>This thesis consists of four independent papers concerningdifferent aspects of interior methods for optimization. Threeof the papers focus on theoretical aspects while the fourth oneconcerns some computational experiments.</p><p>The systems of equations solved within an interior methodapplied to a convex quadratic program can be viewed as weightedlinear least-squares problems. In the first paper, it is shownthat the sequence of solutions to such problems is uniformlybounded. Further, boundedness of the solution to weightedlinear least-squares problems for more general classes ofweight matrices than the one in the convex quadraticprogramming application are obtained as a byproduct.</p><p>In many linesearch interior methods for nonconvex nonlinearprogramming, the iterates can "falsely" converge to theboundary of the region defined by the inequality constraints insuch a way that the search directions do not converge to zero,but the step lengths do. In the sec ond paper, it is shown thatthe multiplier search directions then diverge. Furthermore, thedirection of divergence is characterized in terms of thegradients of the equality constraints along with theasymptotically active inequality constraints.</p><p>The third paper gives a modification of the analytic centerproblem for the set of optimal solutions in linear semidefiniteprogramming. Unlike the normal analytic center problem, thesolution of the modified problem is the limit point of thecentral path, without any strict complementarity assumption.For the strict complementarity case, the modified problem isshown to coincide with the normal analytic center problem,which is known to give a correct characterization of the limitpoint of the central path in that case.</p><p>The final paper describes of some computational experimentsconcerning possibilities of reusing previous information whensolving system of equations arising in interior methods forlinear programming.</p><p><b>Keywords:</b>Interior method, primal-dual interior method,linear programming, quadratic programming, nonlinearprogramming, semidefinite programming, weighted least-squaresproblems, central path.</p><p><b>Mathematics Subject Classification (2000):</b>Primary90C51, 90C22, 65F20, 90C26, 90C05; Secondary 65K05, 90C20,90C25, 90C30.</p>

As unmanned vehicle capabilities have matured, the design and development of autonomous collaborative Systems-of-Systems (SoS) has gained increased attention. This has been motivated by the indication that significant improvements in overall effectiveness may be possible by employing many systems in cooperation with one another. However, as the potential combinations of vehicles, subsystems, and operational concepts becomes increasingly large, a systematic approach is needed for designing and analyzing alternatives. Furthermore, the discrete nature of the problem can cause variations in effectiveness that are counter-intuitive, such as a point of diminishing returns as the number of systems grows. Systems-of-systems are hierarchical in nature, consisting of top-level mission requirements that are decomposed into system- and subsystem-level performance measures. The overarching research objectives of this dissertation are to show that the analysis of alternatives should be performed at varying levels of the SoS hierarchy and to provide novel means for performing those analyses. In particular, it has been postulated that a formulation built on an energy-based approach to multi-level analysis of SoS components will enable more accurate and transparent subsystem and system trade-offs. Various steps of the design process are established and argued for or against, and significant focus is placed on the analysis of alternatives. The foundation of the new method is laid on structured SoS engineering principles. The full substance comes together by incorporating unique aspects developed within this dissertation. A new virtual experimentation approach is presented for creating sensor performance representations that are functions of vehicle operations. The sonar equation is used as a baseline sensor model for comparison against the new virtual experimentation method. Dozens of forward-looking and side-scan sonar experiments are designed, and data is provided to show the extent to which typical sensor modeling over-predicts performance without vehicle operations considered. In addition, comparisons are made between possible representations of vehicle performance. An underwater vehicle sizing and synthesis process is developed to enable comparisons between system-level component modeling approaches. The experiments attest to significant gaps in accuracy when performing sensor and operational trade-offs without energy-based modeling of the collaborative vehicles. Finally, a heuristic path-planning algorithm is formulated, and mixed-integer linear programming is used to choose between alternative SoS designs. The developed method is demonstrated through a representative example problem: a group of unmanned underwater vehicles (UUVs) operating in a collaborative fashion to search for underwater objects. The example scenario provides an application for illustrating the phenomenon discussed in regards to the analysis of alternatives of collaborative SoS. The significance of providing more or less analytic detail is traced and the effect on mission requirements is quantified. Counter-intuitive results are highlighted, such as the observation that the increased energy required for systems to effectively collaborate can often out-weigh the benefits gained in overall mission effectiveness.

Many digital signal processing and real-time streaming systems are modeled using dataflow graphs, such as Synchronous Dataflow (SDF) and Cyclo-static Dataflow (CSDF) graphs that allow static analysis and optimization techniques. However, mapping of such descriptions into tightly constrained real-time implementations requires optimization of resource sharing, buffering and scheduling across a multi-dimensional latency-throughput-area objective space. This requires techniques that can find the Pareto-optimal set of implementations for the designer to choose from. In this work, we address the problem of multi-objective mapping and scheduling of SDF and CSDF graphs onto heterogeneous multi-processor platforms. Building on previous work, this thesis extends existing two-stage hybrid heuristics that combine an evolutionary algorithm with an integer linear programming (ILP) model to jointly optimize throughput, area and latency for SDF graphs. The primary contributions of this work include: (1) extension of the ILP model to support CSDFGs with additional buffer size optimizations; (2) a further optimization in the ILP-based scheduling model to achieve a runtime speedup of almost a factor of 10 compared to the existing SDFG formulation; (3) a list scheduling heuristic that replaces the ILP model in the hybrid heuristic to generate Pareto-optimal solutions at significantly decreased runtime while maintaining near-optimality of the solutions within an acceptable gap of 10% when compared to its ILP counterparts. The list scheduling heuristic presented in this work is based on existing modulo scheduling approaches for software pipelining in the compiler domain, but has been extended by introducing a new concept of mobility-based rescheduling before resorting to backtracking. It has been proved in this work that if mobility-based rescheduling is performed, the number of required backtrackings and hence overall complexity and runtime is less. / text

Στην παρούσα Διπλωματική Εργασία, παρουσιάζεται η πολύ γνωστή μέθοδος Simplex. Με τη βοήθεια της μεθόδου Simplex, μπορούμε να επιλύσουμε προβλήματα γραμμικού προγραμματισμού, ακέραιου γραμμικού προγραμματισμού καθώς και διάφορες παραλλαγές των παραπάνω. Ειδικότερα για τον ακέραιο γραμμικό προγραμματισμό, παρουσιάζονται κάποιες από τις πιο γνωστές μεθόδους αναζήτησης, οι οποίες ανήκουν στην οικογένεια μεθόδων “Branch And Bound”. Επίσης κάποιες τεχνικές αναζήτησης των βέλτιστων λύσεων στο δένδρο που δημιουργείται από τις προηγούμενες τεχνικές. Τα παραπάνω υλοποιήθηκαν σε ένα λογισμικό με γραφικό περιβάλλον (GUI), το οποίο είναι συμβατό με τις περισσότερες εκδόσεις του Λειτουργικού Συστήματος, Windows της Microsoft και χωρίς να χρειάζονται κάτι επιπλέον σε έναν Προσωπικό Υπολογιστή. / This thesis presents the well-known method Simplex. With method Simplex, we can solve problems of linear programming, integer linear programming and several variants of the above. Especially for the integer linear programming, presented some of the most known search methods, which belong to the family of methods "Branch And Bound". Also presented some search techniques for optimal solutions in the tree, generated by the same techniques. These were implemented in a software with graphical interface (GUI), which is compatible with most versions of the Microsoft Windows OS, with a simple installation.

Games on graphs have become an indispensable tool in modern computer science. They provide powerful and expressive models for numerous phenomena and are extensively used in computer- aided verification, automata theory, logic, complexity theory, computational biology, etc. The infinite games on finite graphs we study in this thesis have their primary applications in verification, but are also of fundamental importance from the complexity-theoretic point of view. They include parity, mean payoff, and simple stochastic games. We focus on solving graph games by using iterative strategy improvement and methods from linear programming and combinatorial optimization. To this end we consider old strategy evaluation functions, construct new ones, and show how all of them, due to their structural similarities, fit into a unifying combinatorial framework. This allows us to employ randomized optimization methods from combinatorial linear programming to solve the games in expected subexponential time. We introduce and study the concept of a controlled optimization problem, capturing the essential features of many graph games, and provide sufficent conditions for solvability of such problems in expected subexponential time. The discrete strategy evaluation function for mean payoff games we derive from the new controlled longest-shortest path problem, leads to improvement algorithms that are considerably more efficient than the previously known ones, and also improves the efficiency of algorithms for parity games. We also define the controlled linear programming problem, and show how the games are translated into this setting. Subclasses of the problem, more general than the games considered, are shown to belong to NP intersection coNP, or even to be solvable by subexponential algorithms. Finally, we take the first steps in investigating the fixed-parameter complexity of parity, Rabin, Streett, and Muller games.

In this thesis, a resource allocation problem in OFDMA is studied for the energy efficiency of wireless network. The objective is to minimize the total energy consumption which includes transmission energy consumption, and circuit energy consumption at both transmitter and receiver with required per user’s rate constraint. For problem solution, a heuristic algorithm with low computational complexity and suboptimal solution is proposed, developed in two steps with an increasing order of complexity. Besides, a bounding scheme based on model linearization of formulated nonlinear system model is also proposed to give lower and upper bounds for both small- and large-scale OFDMA network for further algorithm performance evaluation, while the implemented exhaustive search is only capable to provide the optimal solution for small-scale instance for algorithm performance evaluation. Numerical results show that the proposal heuristic algorithm can achieve near-optimal performance with applicable computational complexity even for large-scale networks, and that the bounds from the bounding scheme are very tight for both small- and large-scale OFDMA networks.

Linear programming (LP) decoding of low-density parity-check codes was introduced by Feldman et al. in [1]. In his formulation it is assumed that communication takes place over a memoryless channel and that the source is uniform. Here, we extend the LP decoding paradigm by studying its application to scenarios with source non-uniformity and to decoding over channels with memory. We develop two decoders for the scenario of non-uniform memoryless sources transmitted over memoryless channels. The first decoder uses a modified linear cost function which incorporates the a-priori source information and works with systematic codes. The second decoder differs by using non-systematic codes obtained by puncturing lower rate systematic codes and using an “extended decoding polytope.” Simulations show that the modified decoders yield gains over the standard LP decoder. Next, LP decoding is considered for two channels with memory: the binary additive Markov noise channel and the infinite-memory non-ergodic Polya-contagion channel. For the Markov channel, no linear cost function corresponding to maximum likelihood (ML) decoding could be obtained and hence it is unclear how to proceed. For the Polya channel, two LP-based decoders are developed. The first is derived in a straightforward manner from the ML decoding rule of [2]. The second decoder relies on a simplification of the same ML decoding rule which holds for codes containing the all-ones codeword. Simulations are performed for both decoders with regular and irregular LDPC codes and demonstrate relatively good performance with respect to the channel epsilon-capacity. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2008-12-08 16:24:43.358