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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

A linear program is an optimization problem. In an n dimensional space, whose points are described by variables x1, � , x n, we have a �feasible region� which is a �polytope� by which we mean a region whose boundaries are defined by linear constraints.

A linear program is an optimization problem. In an n dimensional space, whose points are described by variables x1, � , x n, we have a �feasible region� which is a �polytope� by which we mean a region whose boundaries are defined by linear constraints.

A linear program is an optimization problem. In an n dimensional space, whose points are described by variables x1, � , x n, we have a �feasible region� which is a �polytope� by which we mean a region whose boundaries are defined by linear constraints.

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.

Under present arrangements the number of large scale transportation routing problems arising out of SPCC redistribution actions is relatively small. A sample indicates that less than five percent of these problems are of sufficient size and complexity to merit sophisticated computa tional treatment. For these purposes a problem was considered to be sizable if it involved no less than seven activities, no less than three consignees and no less than three consignors. However, it is possible to effect worthwhile savings in transportation outlay on these larger redistribution problems. Even approximative techniques which involve no more complex calcu lations thanhe present rules can save about two percent on transportation mileage in the average redistribution problem. This is a rela tively small figure but considering the large snnual outlay on redistribution transportation cost the absolute dollar saving is likely to be considerable. Most of the calculations were based on straightforward mileage tables rather than transportation cost tables or the proximity tables currently in use by SPCC.

Under present arrangements the number of large scale transportation routing problems arising out of SPCC redistribution actions is relatively small. A sample indicates that less than five percent of these problems are of sufficient size and complexity to merit sophisticated computa tional treatment. For these purposes a problem was considered to be sizable if it involved no less than seven activities, no less than three consignees and no less than three consignors. However, it is possible to effect worthwhile savings in transportation outlay on these larger redistribution problems. Even approximative techniques which involve no more complex calcu lations thanhe present rules can save about two percent on transportation mileage in the average redistribution problem. This is a rela tively small figure but considering the large snnual outlay on redistribution transportation cost the absolute dollar saving is likely to be considerable. Most of the calculations were based on straightforward mileage tables rather than transportation cost tables or the proximity tables currently in use by SPCC.

Under present arrangements the number of large scale transportation routing problems arising out of SPCC redistribution actions is relatively small. A sample indicates that less than five percent of these problems are of sufficient size and complexity to merit sophisticated computa tional treatment. For these purposes a problem was considered to be sizable if it involved no less than seven activities, no less than three consignees and no less than three consignors. However, it is possible to effect worthwhile savings in transportation outlay on these larger redistribution problems. Even approximative techniques which involve no more complex calcu lations thanhe present rules can save about two percent on transportation mileage in the average redistribution problem. This is a rela tively small figure but considering the large snnual outlay on redistribution transportation cost the absolute dollar saving is likely to be considerable. Most of the calculations were based on straightforward mileage tables rather than transportation cost tables or the proximity tables currently in use by SPCC.

Under present arrangements the number of large scale transportation routing problems arising out of SPCC redistribution actions is relatively small. A sample indicates that less than five percent of these problems are of sufficient size and complexity to merit sophisticated computa tional treatment. For these purposes a problem was considered to be sizable if it involved no less than seven activities, no less than three consignees and no less than three consignors. However, it is possible to effect worthwhile savings in transportation outlay on these larger redistribution problems. Even approximative techniques which involve no more complex calcu lations thanhe present rules can save about two percent on transportation mileage in the average redistribution problem. This is a rela tively small figure but considering the large snnual outlay on redistribution transportation cost the absolute dollar saving is likely to be considerable. Most of the calculations were based on straightforward mileage tables rather than transportation cost tables or the proximity tables currently in use by SPCC.

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.

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.

Analogously to classical oriented matroid programming, we give the first feasibility algorithm for signed tropical oriented matroids, which encode the structure of tropical linear inequality systems. The feasibility problem for tropical linear inequality systems is of special interest as it is in NP $\cap$ co-NP but no polynomial time algorithm is known. To encode the combinatorial structure of tropical linear inequality systems, we equip tropical oriented matroids, which where introduced to study the combinatorics of tropical point configurations, with an additional sign information. We generalize the feasibility problem for tropical linear inequality systems to signed tropical oriented matroids. This allows us to formulate pivoting between basic covectors which only depends on the combinatorial structure and not on the coefficients of the inequalities. We combined ideas from tropical geometry, polyhedral combinatorics and combinatorial optimization to deduce a pure graph algorithm from the classical simplex method via its tropicalization. The complexity of the algorithm only depends on the dimension of the input and not on the size of the coefficients. We give an explicit translation between the tropical feasibility problem and the feasibility problem for scheduling with AND-OR-constraints with arbitrary coefficients. Furthermore, we introduce a three-step scheme to transform the general feasibility problem to a better tractable case. We finish by giving a brief explanation of the connection of our result with mean payoff games.

We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument. It is possible to show, interpreting the following notions appropriately, that if a code has a large distance, then its dual has a small covering radius and, therefore, is large. This implies the original code to be small. We also point out (in conjunction with further work) that this bound is a natural isoperimetric constant of the Hamming cube, related to its Faber-Krahn minima. While our approach belongs to the general framework of Delsarte's linear programming method, its main technical ingredient is Fourier duality for the Hamming cube. In particular, we do not deal directly with Delsarte's linear program or orthogonal polynomial theory.

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.

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.

Thesis (M.S. in Operations Research and M.S. in Comp. Sci.)--Naval Postgraduate School, 1979

We discuss the method recently proposed by S. Chubanov for the linear feasibility problem. We present new, concise proofs and interpretations of some of his results. We then show how our proofs can be used to find strongly polynomial time algorithms for special classes of linear feasibility problems. Under certain conditions, these results provide new proofs of classical results obtained by Tardos, and Vavasis and Ye.

We consider semi-infinite linear programs with countably many constraints indexed by the natural numbers. When the constraint space is the vector space of all real valued sequences, we show the finite support (Haar) dual is equivalent to the algebraic Lagrangian dual of the linear program. This settles a question left open by Anderson and Nash [Linear programming in infinite dimensional spaces : theory and applications, Wiley 1987]. This result implies that if there is a duality gap between the primal linear program and its finite support dual, then this duality gap cannot be closed by considering the larger space of dual variables that define the algebraic Lagrangian dual. However, if the constraint space corresponds to certain subspaces of all real-valued sequences, there may be a strictly positive duality gap with the finite support dual, but a zero duality gap with the algebraic Lagrangian dual.

We consider semi-infinite linear programs with countably many constraints indexed by the natural numbers. When the constraint space is the vector space of all real valued sequences, we show the finite support (Haar) dual is equivalent to the algebraic Lagrangian dual of the linear program. This settles a question left open by Anderson and Nash [Linear programming in infinite dimensional spaces : theory and applications, Wiley 1987]. This result implies that if there is a duality gap between the primal linear program and its finite support dual, then this duality gap cannot be closed by considering the larger space of dual variables that define the algebraic Lagrangian dual. However, if the constraint space corresponds to certain subspaces of all real-valued sequences, there may be a strictly positive duality gap with the finite support dual, but a zero duality gap with the algebraic Lagrangian dual.

520 Seiten

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.

Bibliography: l. 132-133

Under some mild Markov assumptions it is shown that the problem of designing optimal sequential tests for two simple hypotheses can be formulated as a linear program. The result is derived by investigating the Lagrangian dual of the sequential testing problem, which is an unconstrained optimal stopping problem, depending on two unknown Lagrangian multipliers. It is shown that the derivative of the optimal cost function with respect to these multipliers coincides with the error probabilities of the corresponding sequential test. This property is used to formulate an optimization problem that is jointly linear in the cost function and the Lagrangian multipliers and an be solved for both with off-the-shelf algorithms. To illustrate the procedure, optimal sequential tests for Gaussian random sequences with different dependency structures are derived, including the Gaussian AR(1) process.

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.

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.

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.

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.

Bibliography: p. 223-226

This paper presents Multi-Objective Integer Non-Linear Programming (MOINLP) involving Negative Total Float (NTF) for improving the basic model of Multi-Objective Programming (MOP) in case the optimization of the additional cost for Project Scheduling Compression (PSC). Using the basic MOP to solve the more complex problems is a challenging task. We suspect that Negative Total Float (NTF) having an indication to make the basic MOP to solve the more general case, both simple and complex of PSC. The purpose of this research is identifying the conflicting objectives in PSC problem using NTF and improving MOINLP by involving the NTF parameter to solve the PSC problem. The Solver Application, which is an add-in of MS Excel, is used to perform optimization process to the model developed. The results show that NTF has an important role to identify the conflicting objectives in PSC. We define NTF is an automatic maximum value of the activity duration reduction to achieve due date of PSC. Furthermore, the use of NTF as a constraint in MOINLP can solve the more general case for both simple and complex PSC problem. Base on the condition, we state that the basic MOP is still significant to solve the PSC complex problems using MOINLP as a sophisticated MOP technique. 

We consider a network of agents whose objective is for the aggregate of their states to converge to a solution of a linear program in standard form. Each agent has limited information about the problem data and can communicate with other agents at discrete time instants of their choosing. Our main contribution is the synthesis of a distributed dynamics and a set of state-based rules, termed triggers, that individual agents use to determine when to opportunistically broadcast their state to neighboring agents to ensure asymptotic convergence to a solution of the linear program. Our technical approach to the algorithm design and analysis overcomes a number of challenges, including establishing convergence in the absence of a common smooth Lyapunov function, ensuring that the triggers are detectable by agents using only local information, accounting for asynchronism in the state broadcasts, and ruling out various causes of arbitrarily fast state broadcasting. Various simulations illustrate our results.

We consider a network of agents whose objective is for the aggregate of their states to converge to a solution of a linear program in standard form. Each agent has limited information about the problem data and can communicate with other agents at discrete time instants of their choosing. Our main contribution is the synthesis of a distributed dynamics and a set of state-based rules, termed triggers, that individual agents use to determine when to opportunistically broadcast their state to neighboring agents to ensure asymptotic convergence to a solution of the linear program. Our technical approach to the algorithm design and analysis overcomes a number of challenges, including establishing convergence in the absence of a common smooth Lyapunov function, ensuring that the triggers are detectable by agents using only local information, accounting for asynchronism in the state broadcasts, and ruling out various causes of arbitrarily fast state broadcasting. Various simulations illustrate our results.

We consider a network of agents whose objective is for the aggregate of their states to converge to a solution of a linear program in standard form. Each agent has limited information about the problem data and can communicate with other agents at discrete time instants of their choosing. Our main contribution is the synthesis of a distributed dynamics and a set of state-based rules, termed triggers, that individual agents use to determine when to opportunistically broadcast their state to neighboring agents to ensure asymptotic convergence to a solution of the linear program. Our technical approach to the algorithm design and analysis overcomes a number of challenges, including establishing convergence in the absence of a common smooth Lyapunov function, ensuring that the triggers are detectable by agents using only local information, accounting for asynchronism in the state broadcasts, and ruling out various causes of arbitrarily fast state broadcasting. Various simulations illustrate our results.

Electrical Engineering 127, 001 - Spring 2014 Creative Commons 3.0: Attribution-NonCommercial-NoDerivs

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

The familiar suboptimal regulator design approach is recast as a constrained optimization problem and incorporated in a Computer Aided Design (CAD) package where both design objective and constraints are quadratic cost functions. This formulation permits the separate consideration of, for example, model following errors, sensitivity measures and control energy as objectives to be minimized or limits to be observed. Efficient techniques for computing the interrelated cost functions and their gradients are utilized in conjunction with a nonlinear programming algorithm. The effectiveness of the approach and the degree of insight into the problem which it affords is illustrated in a helicopter regulation design example.

A time domain technique is developed to design finite-duration impulse response digital filters using linear programming. Two related applications of this technique in data transmission systems are considered. The first is the design of pulse shaping digital filters to generate or detect signaling waveforms transmitted over bandlimited channels that are assumed to have ideal low pass or bandpass characteristics. The second is the design of digital filters to be used as preset equalizers in cascade with channels that have known impulse response characteristics. Example designs are presented which illustrate that excellent waveforms can be generated with frequency-sampling filters and the ease with which digital transversal filters can be designed for preset equalization.

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 $H_1, H_2$.

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.

We examine inhomogeneous primordial nucleosynthesis for {\it arbitrary} distributions $f$ of the baryon-to-photon ratio $\eta$, in the limit where neither particle diffusion nor gravitational collapse is important. By discretizing $f(\eta)$ and using linear programming, we show that for a set of $m$ constraints on the primordial element abundances, the maximum and minimum possible values of $\bar \eta$ (the final mean value of $\eta$) are given when $f(\eta)$ is a sum of at most $m+1$ delta functions. Our linear programming results indicate that when $f$ is taken to be an arbitrary function, there is no lower bound on $\bar \eta$, while the upper bound is essentially the homogeneous upper bound.

This work deals with a class of problems under interval data uncertainty, namely interval robust-hard problems, composed of interval data min-max regret generalizations of classical NP-hard combinatorial problems modeled as 0-1 integer linear programming problems. These problems are more challenging than other interval data min-max regret problems, as solely computing the cost of any feasible solution requires solving an instance of an NP-hard problem. The state-of-the-art exact algorithms in the literature are based on the generation of a possibly exponential number of cuts. As each cut separation involves the resolution of an NP-hard classical optimization problem, the size of the instances that can be solved efficiently is relatively small. To smooth this issue, we present a modeling technique for interval robust-hard problems in the context of a heuristic framework. The heuristic obtains feasible solutions by exploring dual information of a linearly relaxed model associated with the classical optimization problem counterpart. Computational experiments for interval data min-max regret versions of the restricted shortest path problem and the set covering problem show that our heuristic is able to find optimal or near-optimal solutions and also improves the primal bounds obtained by a state-of-the-art exact algorithm and a 2-approximation procedure for interval data min-max regret problems.

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.