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A new method for detecting negative cycles in a graph is proposed. This method is based upon the primal-dual relationships of a linear program formulated from an assignment problem type network. A computer program is developed for this new method to include the complete solution of the assignment problem. Results are given on program efficiency.

Method and Validation for Optimal Lineup Creation for Daily Fantasy Football Using Machine Learning and Linear Programming

Method and Validation for Optimal Lineup Creation for Daily Fantasy Football Using Machine Learning and Linear Programming

A complete computation procedure is given for a special class of two- stage linear programming models in which allocations in the first stage are made to meet an uncertain but known distribution of demands occurring in the second stage. This case, applicable to many practical problems constitutes the principal part of the paper. Next, a class of models is considered where the activities are divided into two or more stages. The quantities of activities in the first stage are the only ones that can be determined in advance because those in the second and latter stages depend on the outcome of random events. Theorems on convexity of the objective (cost) functions are established for the general m-stage case.

A complete computation procedure is given for a special class of two- stage linear programming models in which allocations in the first stage are made to meet an uncertain but known distribution of demands occurring in the second stage. This case, applicable to many practical problems constitutes the principal part of the paper. Next, a class of models is considered where the activities are divided into two or more stages. The quantities of activities in the first stage are the only ones that can be determined in advance because those in the second and latter stages depend on the outcome of random events. Theorems on convexity of the objective (cost) functions are established for the general m-stage case.

A set of linear programming test problems is analyzed with MINOS, Version 5.1. The problems have been run with different options for scaling and partial pricing to illustrate the effects of these options on the performance of the simplex method. The results indicate that the different options can significantly improve or degrade the performance of the simplex method, and that these options must be chosen wisely. For each problem, a picture of the nonzero structure of the matrix A is also presented so that the problems can be classified according to structure. (Author)

We proposed a build-up interior method for solving an m equation n variable linear program which has the same convergence properties as their well known analogues in dual affine and projective forms but requires less computational effort. The algorithm has three forms, an affine scaling form, a projective scaling form, and an exact form (that used pivot steps). In this paper, we present the first of these. It differs from Dikin's algorithm of dual affine form in that the ellipsoid chosen to generate the improving directions in dual space is constructed from only a subset of the dual constraints. (KR)

We proposed a build-up interior method for solving an m equation n variable linear program which has the same convergence properties as their well known analogues in dual affine and projective forms but requires less computational effort. The algorithm has three forms, an affine scaling form, a projective scaling form, and an exact form (that used pivot steps). In this paper, we present the first of these. It differs from Dikin's algorithm of dual affine form in that the ellipsoid chosen to generate the improving directions in dual space is constructed from only a subset of the dual constraints. (KR)

We proposed a build-up interior method for solving an m equation n variable linear program which has the same convergence properties as their well known analogues in dual affine and projective forms but requires less computational effort. The algorithm has three forms, an affine scaling form, a projective scaling form, and an exact form (that used pivot steps). In this paper, we present the first of these. It differs from Dikin's algorithm of dual affine form in that the ellipsoid chosen to generate the improving directions in dual space is constructed from only a subset of the dual constraints. (KR)

We proposed a build-up interior method for solving an m equation n variable linear program which has the same convergence properties as their well known analogues in dual affine and projective forms but requires less computational effort. The algorithm has three forms, an affine scaling form, a projective scaling form, and an exact form (that used pivot steps). In this paper, we present the first of these. It differs from Dikin's algorithm of dual affine form in that the ellipsoid chosen to generate the improving directions in dual space is constructed from only a subset of the dual constraints. (KR)

Given a point sufficiently close to a nondegenerate basic feasible solution x* of a linear program, we show how to generate a sequence {p-superscript-k} that converges to the 0-1 vector sign(x*) at a Q-cubic rate. This extremely fast convergence enables us to determine, with a high degree of certainty, which variables will be zero and which will be nonzero at optimality and then construct x* from this information.

This article considers the problem of bounding the expected value of a linear program (LP) containing random coefficients, with applications to solving two-stage stochastic programs. An upper bound for minimizations is derived from a restriction of an equivalent, penalty-based formulation of the primal stochastic LP, and a lower bound is obtained from a restriction of a reformulation of the dual. These restricted-recourse bounds are more general and more easily computed than most other bounds because random coefficients may appear anywhere in the LP, neither independence nor boundedness of the coefficients is needed, and the bound is computed by solving a single LP or nonlinear program. Analytical examples demonstrate that the new bounds can be stronger than complementary Jensen bounds. (An upper bound is complementary to a lower bound, and vice versa). In computational work, the authors apply the bounds to a two-stage stochastic program for semiconductor manufacturing with uncertain demand and production rates.

This article considers the problem of bounding the expected value of a linear program (LP) containing random coefficients, with applications to solving two-stage stochastic programs. An upper bound for minimizations is derived from a restriction of an equivalent, penalty-based formulation of the primal stochastic LP, and a lower bound is obtained from a restriction of a reformulation of the dual. These restricted-recourse bounds are more general and more easily computed than most other bounds because random coefficients may appear anywhere in the LP, neither independence nor boundedness of the coefficients is needed, and the bound is computed by solving a single LP or nonlinear program. Analytical examples demonstrate that the new bounds can be stronger than complementary Jensen bounds. (An upper bound is complementary to a lower bound, and vice versa). In computational work, the authors apply the bounds to a two-stage stochastic program for semiconductor manufacturing with uncertain demand and production rates.

Given a collection of discrete random variables representing outcomes of learned local predictors in natural language. e.g.. named entities and relations. we seek an optimal global assignment to the variables in the presence of general (non-sequential) constraints. Examples of these constraints include the type of arguments a relation can take, and the mutual activity of different relations. etc. We develop a linear programing formulation for this problem and evaluate it in the context of simultaneously learning named entities and relations. Our approach allows us to efficiently incorporate domain and task specific constraints at decision time, resulting in significant improvements in the accuracy and the human-like quality of the inferences.

In future conflicts, the projected expenditure rates of artillery ammunition greatly exceed the ability of the Marine direct support artillery battery's ammunition transportation assets. It is therefore vital that the artillery battery commander be able to select the most effective mix of ammunition to carry on his organic transportation in a given tactical situation. Linear programing is a tool which the batten' commander can use to help solve this important problem. This thesis provides a linear program to assist him in this solution. In addition, with slight modification, this linear program can be of use to commanders at all levels of the Marine artillerv organization.

In this paper, the linear programming (LP) decoder for binary linear codes, introduced by Feldman, et al. is extended to joint-decoding of binary-input finite-state channels. In particular, we provide a rigorous definition of LP joint-decoding pseudo-codewords (JD-PCWs) that enables evaluation of the pairwise error probability between codewords and JD-PCWs. This leads naturally to a provable upper bound on decoder failure probability. If the channel is a finite-state intersymbol interference channel, then the LP joint decoder also has the maximum-likelihood (ML) certificate property and all integer valued solutions are codewords. In this case, the performance loss relative to ML decoding can be explained completely by fractional valued JD-PCWs.

The motivation for this work has been the need for a practical procedure to solve the maximum-weight cut problem (MCP) in undirected graphs. Our primary focus has been on problems arising from considerations in statistical mechanics. These problems are typically posed on grid graphs and some natural variants. There has been significant progress in two areas: solution of the maximum-weight cut problem and development of simplex-based tools for integer programming. Codes developed have been widely used to improve solution time.

This brief paper: (1) Discusses strategies to generate random test cases that can be used to extensively test any Linear Distance Program (LDP) software. (2) Gives three numerical examples of input cases generated by this strategy that cause problems in the Lawson and Hanson LDP module. (3) Proposes, as a standard matter of acceptable implementation procedures, that (unless it is done internally in the software itself, but, in general, this seems to be much rarer than one would expect) all users should test the returned output from any LDP module for self-consistency since it incurs only a small amount of added computational overhead and it is not hard to do.

An algorithm for an approximating function to the frequency distribution is obtained from a sample of size n. To obtain the approximating function a histogram is made from the data. Next, Euclidean space approximations to the graph of the histogram using central B-splines as basis elements are obtained by linear programming. The approximating function has area one and is nonnegative.

The linear programing approach to position-salary evaluation of school district personnel allows the district to calculate a system of relative weights that establish the relationship of one position to another in quantitative terms. The model provides an internally consistent, effective assessment of an individual's relative worth to the school district in terms of salary. Linear programing models provide fairly sophisticated methods for efficient allocations of resources consistent with the objectives and constraints of the school district. A related document is EA 003 594. (Author/RA)

The linear programing approach to position-salary evaluation of school district personnel allows the district to calculate a system of relative weights that establish the relationship of one position to another in quantitative terms. The model provides an internally consistent, effective assessment of an individual's relative worth to the school district in terms of salary. Linear programing models provide fairly sophisticated methods for efficient allocations of resources consistent with the objectives and constraints of the school district. A related document is EA 003 594. (Author/RA)

The linear programing approach to position-salary evaluation of school district personnel allows the district to calculate a system of relative weights that establish the relationship of one position to another in quantitative terms. The model provides an internally consistent, effective assessment of an individual's relative worth to the school district in terms of salary. Linear programing models provide fairly sophisticated methods for efficient allocations of resources consistent with the objectives and constraints of the school district. A related document is EA 003 594. (Author/RA)

A nonlinear programming technique was developed for the automated design and optimization of autopilots for large flexible launch vehicles. This technique, which resulted in the COEBRA program, uses the iterative application of linear programming. The method deals directly with the three main requirements of booster autopilot design: to provide (1) good response to guidance commands; (2) response to external disturbances (e.g. wind) to minimize structural bending moment loads and trajectory dispersions; and (3) stability with specified tolerances on the vehicle and flight control system parameters. The method is applicable to very high order systems (30th and greater per flight condition). Examples are provided that demonstrate the successful application of the employed algorithm to the design of autopilots for both single and multiple flight conditions.

Much research has centered on the problem of finding shortest paths in graphs. It is well known that there is a direct correspondence between the single-source shortest-paths problem and the following simple linear programming problem. Let S be a set of linear inequalities of the form x sub j - x sub i or - a sub ij, where the x sub i are unknowns and the a sub ij are given rea constants. Determine a set of values for the x sub i such that the inequalities in S are satisfied, or determine that no such values exist. This paper considers the mixed-integer linear programming variant of this problem in which some (but not necessarily all) of the x sub i are required to be integers. The problem arises in the context of synchronous circuit optimization, but it has applications to PERT scheduling and VLSI layout compaction as well.

Annual decisions concerning procurement of aircraft, munitions, and spare parts are of enormous interest to senior military decisionmakers. Currently the Air Force uses the Theater Attack Model (TAM), a large-scale linear program, to evaluate theater level tactical air operations in support of procurement decisions. The Air Force Center for Studies and Analysis (AFCSA) maintains TAM and uses it to conduct trade-off analyses by examining the impact of the following factors: budget changes; aircraft and munitions effectiveness; target values; attrition rates; the costs of current and forecast aircraft, munitions, and spares; existing force structure of aircraft and munitions; weather; length of mission; and length of conflict. Currently, AFCSA is examining ways to expand TAM to include airbase operability and electronic countermeasures. The Theater Attack Model (TAM) is a large-scale linear program (LP) used to aid senior decisionmakers in making the tough budget and procurement decisions for the Air Force. TAM, as currently configured, can generate matrices with 9 million variables. The CPU-time to run this model is enormous. Advanced LP techniques are examined to reduce TAM's CPU-time. Recommendations for reducing TAM's CPU-time are outlined with the hope of saving the government money in computer time. Keywords: Simplex method, Mathematical models, Karmarkar's algorithm.

In the summer of 2005, a RAND Corporation study team was asked to assist the Air Force in assessing Air Force culture and its relationship to a range of behaviors it deemed aberrant. We developed a questionnaire for a survey of Air Force personnel on cultural attitudes (henceforth called the CULTURE survey), and designed a sample of the population to receive email invitations to participate in the survey. The design needed to meet a number of goals that may concern other survey researchers as well: (1) minimize the number of people asked to participate so as to reduce the survey burden on a population already frequently invited to take surveys; (2) reflect response rates we could anticipate from previous surveys of the population; (3) ensure adequate representation of a number of minorities of interest (rank, job type, race and ethnicity, gender, religion, and component); (4) sample enough people in each of the overlapping subset categories of interest (e.g., black female noncommissioned officers [NCOs]) to allow for statistically meaningful comparisons; and (5) minimize (to zero, if possible) the number of service members invited to take both this survey and another survey (the HEALTH survey) on an overlapping set of topics scheduled for the same time period. We describe here the method we developed for designing joint samples for the CULTURE and HEALTH surveys. The Air Force personnel inventory consists of approximately 350,000 active, 105,000 Air National Guard, 75,000 Air Force Reserve, and 150,000 civilian personnel. While our survey drew samples from all these groups, in this report we illustrate the method for the Guard and Reserve only. We wish to select a sample of these individuals that is large and diverse enough to allow us to draw conclusions about how their attitudes are related to their various personal and professional characteristics.

In the summer of 2005, a RAND Corporation study team was asked to assist the Air Force in assessing Air Force culture and its relationship to a range of behaviors it deemed aberrant. We developed a questionnaire for a survey of Air Force personnel on cultural attitudes (henceforth called the CULTURE survey), and designed a sample of the population to receive email invitations to participate in the survey. The design needed to meet a number of goals that may concern other survey researchers as well: (1) minimize the number of people asked to participate so as to reduce the survey burden on a population already frequently invited to take surveys; (2) reflect response rates we could anticipate from previous surveys of the population; (3) ensure adequate representation of a number of minorities of interest (rank, job type, race and ethnicity, gender, religion, and component); (4) sample enough people in each of the overlapping subset categories of interest (e.g., black female noncommissioned officers [NCOs]) to allow for statistically meaningful comparisons; and (5) minimize (to zero, if possible) the number of service members invited to take both this survey and another survey (the HEALTH survey) on an overlapping set of topics scheduled for the same time period. We describe here the method we developed for designing joint samples for the CULTURE and HEALTH surveys. The Air Force personnel inventory consists of approximately 350,000 active, 105,000 Air National Guard, 75,000 Air Force Reserve, and 150,000 civilian personnel. While our survey drew samples from all these groups, in this report we illustrate the method for the Guard and Reserve only. We wish to select a sample of these individuals that is large and diverse enough to allow us to draw conclusions about how their attitudes are related to their various personal and professional characteristics.

In the summer of 2005, a RAND Corporation study team was asked to assist the Air Force in assessing Air Force culture and its relationship to a range of behaviors it deemed aberrant. We developed a questionnaire for a survey of Air Force personnel on cultural attitudes (henceforth called the CULTURE survey), and designed a sample of the population to receive email invitations to participate in the survey. The design needed to meet a number of goals that may concern other survey researchers as well: (1) minimize the number of people asked to participate so as to reduce the survey burden on a population already frequently invited to take surveys; (2) reflect response rates we could anticipate from previous surveys of the population; (3) ensure adequate representation of a number of minorities of interest (rank, job type, race and ethnicity, gender, religion, and component); (4) sample enough people in each of the overlapping subset categories of interest (e.g., black female noncommissioned officers [NCOs]) to allow for statistically meaningful comparisons; and (5) minimize (to zero, if possible) the number of service members invited to take both this survey and another survey (the HEALTH survey) on an overlapping set of topics scheduled for the same time period. We describe here the method we developed for designing joint samples for the CULTURE and HEALTH surveys. The Air Force personnel inventory consists of approximately 350,000 active, 105,000 Air National Guard, 75,000 Air Force Reserve, and 150,000 civilian personnel. While our survey drew samples from all these groups, in this report we illustrate the method for the Guard and Reserve only. We wish to select a sample of these individuals that is large and diverse enough to allow us to draw conclusions about how their attitudes are related to their various personal and professional characteristics.

In the summer of 2005, a RAND Corporation study team was asked to assist the Air Force in assessing Air Force culture and its relationship to a range of behaviors it deemed aberrant. We developed a questionnaire for a survey of Air Force personnel on cultural attitudes (henceforth called the CULTURE survey), and designed a sample of the population to receive email invitations to participate in the survey. The design needed to meet a number of goals that may concern other survey researchers as well: (1) minimize the number of people asked to participate so as to reduce the survey burden on a population already frequently invited to take surveys; (2) reflect response rates we could anticipate from previous surveys of the population; (3) ensure adequate representation of a number of minorities of interest (rank, job type, race and ethnicity, gender, religion, and component); (4) sample enough people in each of the overlapping subset categories of interest (e.g., black female noncommissioned officers [NCOs]) to allow for statistically meaningful comparisons; and (5) minimize (to zero, if possible) the number of service members invited to take both this survey and another survey (the HEALTH survey) on an overlapping set of topics scheduled for the same time period. We describe here the method we developed for designing joint samples for the CULTURE and HEALTH surveys. The Air Force personnel inventory consists of approximately 350,000 active, 105,000 Air National Guard, 75,000 Air Force Reserve, and 150,000 civilian personnel. While our survey drew samples from all these groups, in this report we illustrate the method for the Guard and Reserve only. We wish to select a sample of these individuals that is large and diverse enough to allow us to draw conclusions about how their attitudes are related to their various personal and professional characteristics.

Since 1988, the United States Army has closed 112 and has completed or will soon complete realignment of another 27 of its domestic installations. The Army estimates the total cost (between 1988 and 2001) of these closures and realignments to be $5.3 billion, of which about $2.3 billion (43%) is associated with environmental cleanup. Beyond 2001, the Army expects to spend an additional $ 1.09 billion to complete cleanup and continue restoration. The Army Base Realignment and Closure Office (BRACO) is currently funding environmental cleanup at 649 sites on 39 current and former Army installations. BRACO' 5 environmental restoration budget from 2001 to 2007 to support cleanup at these installations (totaling over $620 million) is not sufficient to support each installation's requirement for those years. Considering environmental policies and yearly funding requests from 2001 to 2015 for each site, this thesis develops optimization models and a spreadsheet interface to help BRACO allocate its budget. Model results prescribe either funding each site as requested or delaying cleanup by one to five years. Extensive model use helped BRACO analyze alternate yearly budgets, suggest alternate site funding and determine site funding for 2001 to 2007

ADA001247

We show how categorial deduction can be implemented in higher-order (linear) logic programming, thereby realising parsing as deduction for the associative and non-associative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variety of its extensions.

An introduction to Dantzig's simplex method of finding optimal solutions.

We study the intersection of finitely generated factor-free subgroups of free products of groups by utilizing the method of linear programming. For example, we prove that if $H_1$ is a finitely generated factor-free noncyclic subgroup of the free product $G_1 * G_2$ of two finite groups $G_1$, $G_2$, then the Walter Neumann coefficient $\sigma(H_1)$ of $H_1$ is rational and can be computed. This coefficient $\sigma(H_1)$ is the minimal positive real number such that, for every finitely generated factor-free subgroup $H_2$ of $G_1 * G_2$, it is true that $\bar {\rm r}(H_1, H_2) \le \sigma(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2)$, where $\bar {\rm r} (H) = \max ( {\rm r} (H)-1,0)$ is reduced rank of $H$, ${\rm r}(H)$ is rank of $H$, and $\bar {\rm r}(H_1, H_2)$ is reduced rank of a generalized intersection of $H_1, H_2$.

We study the intersection of finitely generated factor-free subgroups of free products of groups by utilizing the method of linear programming. For example, we prove that if $H_1$ is a finitely generated factor-free noncyclic subgroup of the free product $G_1 * G_2$ of two finite groups $G_1$, $G_2$, then the Walter Neumann coefficient $\sigma(H_1)$ of $H_1$ is rational and can be computed. This coefficient $\sigma(H_1)$ is the minimal positive real number such that, for every finitely generated factor-free subgroup $H_2$ of $G_1 * G_2$, it is true that $\bar {\rm r}(H_1, H_2) \le \sigma(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2)$, where $\bar {\rm r} (H) = \max ( {\rm r} (H)-1,0)$ is reduced rank of $H$, ${\rm r}(H)$ is rank of $H$, and $\bar {\rm r}(H_1, H_2)$ is reduced rank of a generalized intersection of $H_1, H_2$.

An algorithm is described for determining the optimal solution of parametric linear and quadratic programming problems as an explicit piecewise linear function of the parameter. Each linear function is uniquely determined by an appropriate subset of active constraints. For every critical value of the parameter a new subset has to be determined. A simple rule is given for adding and deleting constraints from this subset. (Author)

An algorithm is described for determining the optimal solution of parametric linear and quadratic programming problems as an explicit piecewise linear function of the parameter. Each linear function is uniquely determined by an appropriate subset of active constraints. For every critical value of the parameter a new subset has to be determined. A simple rule is given for adding and deleting constraints from this subset. (Author)

An algorithm is described for determining the optimal solution of parametric linear and quadratic programming problems as an explicit piecewise linear function of the parameter. Each linear function is uniquely determined by an appropriate subset of active constraints. For every critical value of the parameter a new subset has to be determined. A simple rule is given for adding and deleting constraints from this subset. (Author)

The main purpose of this work is to give explicit sparsity-preserving SOR(Successive Overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming. (Author)

The performance of two different approaches to solving the non-linear control allocation problem is presented. The non-linear control allocation problem is formulated using piecewise linear functions to approximate the control moments produced by a set of control effectors. when the control allocation problem is formulated as a piecewise linear program, an additional set of constraints enter into the problem formation. One approach is to introduce a set of binary variables to enforce these constraints. The result is a mixed- integer linear programming problem that can be solved using any branch-and-bound software. A second approach is to solve the piecewise linear programming problem using a modified simplex method. The simplex algorithm is modified to enforce a subset of the decision variables to enter into the basis only if certain conditions are met. We will show that solving the optimization problem using the simplex based approach is significantly faster than solving the same problem using a mixed-integer formulation. We will then compare the closed-loop performance of a re-entry vehicle using both approaches.

An algorithm to solve linear programming problems is presented in this thesis which is based on Karmarkar's projective method. The algorithm includes a practical method to project a general linear programming problem onto a unit simplex and eliminates the a priori need to know the optimal value of the objective function. The implementation conserves sparsity. The key part of the implementation is the solution of a linear least-squares problem to find an improving direction: a direct and an iterative method are implemented to solve this problem. The direct method employs the minimum-degree heuristic to reorder the system of normal equations, and thus conserve sparsity during the following Cholesky factor of the normal equation matrix as a preconditioner for conjugate gradient iterations which are performed implicity on the preconditioned matrix. The study concludes with implementation remarks, and computational results.

Since 1988, the United States Army has closed 112 and has completed or will soon complete realignment of another 27 of its domestic installations. The Army estimates the total cost (between 1988 and 2001) of these closures and realignments to be $5.3 billion, of which about $2.3 billion (43%) is associated with environmental cleanup. Beyond 2001, the Army expects to spend an additional $ 1.09 billion to complete cleanup and continue restoration. The Army Base Realignment and Closure Office (BRACO) is currently funding environmental cleanup at 649 sites on 39 current and former Army installations. BRACO' 5 environmental restoration budget from 2001 to 2007 to support cleanup at these installations (totaling over $620 million) is not sufficient to support each installation's requirement for those years. Considering environmental policies and yearly funding requests from 2001 to 2015 for each site, this thesis develops optimization models and a spreadsheet interface to help BRACO allocate its budget. Model results prescribe either funding each site as requested or delaying cleanup by one to five years. Extensive model use helped BRACO analyze alternate yearly budgets, suggest alternate site funding and determine site funding for 2001 to 2007.

This report examines the role played by several kinds of ceiling points in solving the pure, general integer linear programming problem (ILP). While no assumptions are made concerning the structure or signs of the data of the problem, it is assumed that the feasible region for (ILP) is non-empty and bounded. A ceiling point with respect to a single constraint maybe thought of as an integer solution on or close to the boundary of the feasible region defined by the constraint. The definition of a ceiling point with respect to a single constraint is extended to take multiple constraints into consideration simultaneously, defining what is called a feasible ceiling point. It is shown that the set all feasible ceiling points contains at least one optimal solution for (ILP). A related class of solutions called feasible 1-ceiling points is also characterized and shown to contain all optimal solutions for (ILP). Moreover, 1- ceiling points are computationally easier to identify than ordinary ceiling points and may be sought with respect to one constant at a time. It is also demonstrated that solving (ILP) requires only enumerating feasible 1-ceiling points with respect to a subset of all functional constraints. Keywords: Integer variables; Enumeration algorithms.

Considering the intermittency of renewable energy systems, a sizing and scheduling model is proposed for a finite number of static electric loads. The model objective is to maximize solar energy utilization with and without storage. For the application of optimal load size selection, the energy production of a solar photovoltaic is assumed to be consumed by a finite number of discrete loads in an off-grid system using mixed-integer linear programming. Additional constraints are battery charge and discharge limitations and minimum uptime and downtime for each unit. For a certain solar power profile the model outputs optimal unit size as well as the optimal scheduling for both units and battery charge and discharge (if applicable). The impact of different solar power profiles and minimum up and down time constraints on the optimal unit and battery sizes are studied. The battery size required to achieve full solar energy utilization decreases with the number of units and with increased flexibility of the units (shorter on and off-time). A novel formulation is introduced to model quasi-dynamic units that gradually start and stop and the quasi-dynamic units increase solar energy utilization. The model can also be applied to search for the optimal number of units for a given cost function.

We consider the problem of the computation of $\inf_p \theta p$ over the set of exponent pairs $P \ni p$ under linear constraints for a certain class of objective functions $\theta$. An effective algorithm is presented. The output of the algorithm leads to the improvement and establishing new estimates in the various divisor problems in the analytic number theory.

In the paper we consider the linear underdetermined system of a special type. Systems of this type appear in non-homogeneous network flow programming problems in the form of systems of constraints and can be characterized as systems with a large sparse submatrix representing the embedded network structure. We develop a direct method for finding solutions of the system. The algorithm is based on the theoretic-graph specificities for the structure of the support and properties of the basis of a solution space of a homogeneous system. One of the key steps is decomposition of the system. A simple example is regarded at the end of the paper.

We introduce a linear programming method to obtain bounds on the cardinality of codes in Grassmannian spaces for the chordal distance. We obtain explicit bounds, and an asymptotic bound that improves on the Hamming bound. Our approach generalizes the approach originally developed by P. Delsarte and Kabatianski-Levenshtein for compact two-point homogeneous spaces.

We consider the problem of tracking a target whose dynamics is modeled by a continuous It\=o semi-martingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds for this problem, depending on the cost structure. These lower bounds can be related to the time-average control of Brownian motion, which is characterized as a deterministic linear programming problem. A comprehensive list of examples with explicit expressions for the lower bounds is provided.

We introduce a new approach for the numerical pricing of American options. The main idea is to choose a finite number of suitable excessive functions (randomly) and to find the smallest majorant of the gain function in the span of these functions. The resulting problem is a linear semi-infinite programming problem, that can be solved using standard algorithms. This leads to good upper bounds for the original problem. For our algorithms no discretization of space and time and no simulation is necessary. Furthermore it is applicable even for high-dimensional problems. The algorithm provides an approximation of the value not only for one starting point, but for the complete value function on the continuation set, so that the optimal exercise region and e.g. the Greeks can be calculated. We apply the algorithm to (one- and) multidimensional diffusions and to L\'evy processes, and show it to be fast and accurate.

The main purpose of this work is to give explicit sparsity-preserving SOR(Successive Overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming. (Author)