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This is a representation of a linear optimization problem, also known as a linear program. This shape corresponds to an example known as the "Brewery Problem". Its objective is to maximize profit from the sale of three beers (light, dark, and ale), each requiring different combinations of three limited resources (barley, hops, and yeast). The polytope represents the feasible region, that is, amounts of the three beers that can be brewed without exhausting the resources. The plane represents the profit. Every point on that plane has the same level of profit. The optimal solution to the problem is where the plane intersects the polytope. There is more to it (see for example this post ) but that is the basic idea.

This is a representation of a linear optimization problem, also known as a linear program. This shape corresponds to an example known as the "Brewery Problem". Its objective is to maximize profit from the sale of three beers (light, dark, and ale), each requiring different combinations of three limited resources (barley, hops, and yeast). The polytope represents the feasible region, that is, amounts of the three beers that can be brewed without exhausting the resources. The plane represents the profit. Every point on that plane has the same level of profit. The optimal solution to the problem is where the plane intersects the polytope. There is more to it (see for example this post ) but that is the basic idea.

Bibliography: p. 201-203

Bibliography: p. 201-203

STEP FSC 2ND YEAR PART II XII MATHEMATICS LECTURES, DO NOT SELL!!, FREE FOR STUDIOUS STUDENTS. ALL COPYRIGHTS BELONGS TO STEP BY PGC

Signal to Noise Ratio (SNR) is an important index for wireless communications. In CDMA systems, spreading sequences are utilized. This series of papers show the method to derive spreading sequences as the solutions of the non-linear programming: maximize SNR. In this paper, we consider a frequency-selective wide-sense-stationary uncorrelated-scattering (WSSUS) channel and evaluate the worst case of SNR. Then, we derive the new expression of SNR whose main term consists of the periodic correlation terms and the aperiodic correlation terms. In general, there is a relation between SNR and mean-square correlations, which are indices for performance of spreading sequences. Then, we show the relation between our expression and them. With this expression, we can maximize SNR with the Lagrange multiplier method. In Part II, with this expression, we construct two types optimization problems and evaluate them.

Signal to Noise Ratio (SNR) is an important index for wireless communications. In CDMA systems, spreading sequences are utilized. This series of papers show the method to derive spreading sequences as the solutions of the non-linear programming: maximize SNR. In this paper, we consider a frequency-selective wide-sense-stationary uncorrelated-scattering (WSSUS) channel and evaluate the worst case of SNR. Then, we derive the new expression of SNR whose main term consists of the periodic correlation terms and the aperiodic correlation terms. In general, there is a relation between SNR and mean-square correlations, which are indices for performance of spreading sequences. Then, we show the relation between our expression and them. With this expression, we can maximize SNR with the Lagrange multiplier method. In Part II, with this expression, we construct two types optimization problems and evaluate them.

Thesis (MS)?Naval Postgraduate School, 1967

Thesis (MS)?Naval Postgraduate School, 1967

Learning and retention may be influenced by subtle instructional stimulus characteristics and certain visual memory aptitudes. Ten stimulus characteristics were chosen for study; 50 sequences of programed instructional material were specially written to conform to sampled values of each stimulus characteristic. Seventy-three freshman subjects received the 50 sequences and then took an immediate and a delayed (one-week) posttest to assess learning. Measures on four visual memory and cognition aptitude factors were available for 43 of the subjects. By means of tear-down regression algorithms, the 10 stimulus characteristics were used to predict to the learning criteria. The ratio of examples within a sequence to the number of frames in the sequence had a correlation of about -.70 with both the immediate and delayed posttest. The inclusion of other variables did not increase the prediction significantly. Together, total frames and number of responses per frame predicted item difficulty on the posttest (multiple R to the second power=.90). Of the aptitude variables, Vocabulary aptitude and Short Term Object Memory tended to increase prediction to the delayed posttest criterion, while Serial Integration aptitude and short Term Color Memory did not. (LS)

Scan of book Linear programming and applications, a course text by Will McLewin [1980] {519.72--oclc} Content material:- Conversion to specified form: Basic, Feasible and Optimum solutions The Simplex Method Duality Parametric linear programming and sensitivity analysis The Shor-Khachian ellipsoid method Transportation and similar problems Network flows The marriage problem Games theory: two person matrix games Quadratic programming Functional aproximation, Matrix eigenvalue perturbation analysis

What is the value of information in solving linear programming? The celebrated ellipsoid algorithm tells us that the full information of input constraints is not necessary; the algorithm works as long as there exists an oracle that, on a proposed candidate solution, returns a violation in the format of a separating hyperplane. Can linear programming still be efficiently solved if the returned violation is in another format? We study this question in a trial-and-error setting: there is an oracle that, upon a proposed solution, returns the index of an arbitrary violated constraint (with the content of the constraint still hidden). We give an algorithm with running time O(m^{poly(n)} L), where m and n are the numbers of constraints and variables, respectively, and L is the input size of the linear program. The exponential dependence on n is unfortunately unavoidable; we show a lower bound of \Omega(m^{\lfloor n/2\rfloor}) on the number of queries needed. Meanwhile, if the oracle provides more violation information---the index of a "most violated" constraint, measured by the Euclidean distance of the proposed solution and the half-spaces defined by the constraints---then we show that the linear program can be solved in polynomial time. The proofs of the results employ a variety of geometric techniques, including McMullen's Upper Bound Theorem, the weighted spherical Voronoi diagram, and the furthest Voronoi diagram. In addition, we give an alternative proof to a conjecture of L\'aszl\'o Fejes T\'oth on bounding the number of disconnected components formed by the union of m convex bodies in R^n. Our proof, inspired by the Gauss-Bonnet Theorem in global differential geometry, is independent of the known and clearly reveals more insights into the problem and bound.

We are concerned with the linear-quadratic optimal stochastic control problem with random coefficients. Under suitable conditions, we prove that the value field $V(t,x,\omega), (t,x,\omega)\in [0,T]\times R^n\times \Omega$, is quadratic in $x$, and has the following form: $V(t,x)=\langle K_tx, x\rangle$ where $K$ is an essentially bounded nonnegative symmetric matrix-valued adapted processes. Using the dynamic programming principle (DPP), we prove that $K$ is a continuous semi-martingale of the form $$K_t=K_0+\int_0^t \, dk_s+\sum_{i=1}^d\int_0^tL_s^i\, dW_s^i, \quad t\in [0,T]$$ with $k$ being a continuous process of bounded variation and $$E\left[\left(\int_0^T|L_s|^2\, ds\right)^p\right]

In this paper, we introduce an efficient iterative solver for the joint linear-programming (LP) decoding of low-density parity-check (LDPC) codes and finite-state channels (FSCs). In particular, we extend the approach of iterative approximate LP decoding, proposed by Vontobel and Koetter and explored by Burshtein, to this problem. By taking advantage of the dual-domain structure of the joint decoding LP, we obtain a convergent iterative algorithm for joint LP decoding whose structure is similar to BCJR-based turbo equalization (TE). The result is a joint iterative decoder whose complexity is similar to TE but whose performance is similar to joint LP decoding. The main advantage of this decoder is that it appears to provide the predictability of joint LP decoding and superior performance with the computational complexity of TE.

In this paper, we introduce an efficient iterative solver for the joint linear-programming (LP) decoding of low-density parity-check (LDPC) codes and finite-state channels (FSCs). In particular, we extend the approach of iterative approximate LP decoding, proposed by Vontobel and Koetter and explored by Burshtein, to this problem. By taking advantage of the dual-domain structure of the joint decoding LP, we obtain a convergent iterative algorithm for joint LP decoding whose structure is similar to BCJR-based turbo equalization (TE). The result is a joint iterative decoder whose complexity is similar to TE but whose performance is similar to joint LP decoding. The main advantage of this decoder is that it appears to provide the predictability of joint LP decoding and superior performance with the computational complexity of TE.

A class of non-convex optimization problems with DC objective function is studied, where DC stands for being representable as the difference $f=g-h$ of two convex functions $g$ and $h$. In particular, we deal with the special case where one of the two convex functions $g$ or $h$ is polyhedral. In case $g$ is polyhedral, we show that a solution of the DC program can be obtained from a solution of an associated polyhedral projection problem. In case $h$ is polyhedral, we prove that a solution of the DC program can be obtained by solving a polyhedral projection problem and finitely many convex programs. Since polyhedral projection is equivalent to multiple objective linear programming (MOLP), a MOLP solver (in the second case together with a convex programming solver) can be used to solve instances of DC programs with polyhedral component. Numerical examples are provided, among them an application to locational analysis.

We consider a system of nonlinear ordinary differential equations for the solution of linear programming (LP) problems that was first proposed in the mathematical biology literature as a model for the foraging behavior of acellular slime mold Physarum polycephalum, and more recently considered as a method to solve LPs. We study the convergence time of the continuous Physarum dynamics in the context of the linear programming problem, and derive a new time bound to approximate optimality that depends on the relative entropy between projected versions of the optimal point and of the initial point. The bound scales logarithmically with the LP cost coefficients and linearly with the inverse of the relative accuracy, establishing the efficiency of the dynamics for arbitrary LP instances with positive costs.

Markov Decision Processes (MDP) is an useful framework to cast optimal sequential decision making problems. Given any MDP the aim is to find the optimal action selection mechanism i.e., the optimal policy. Typically, the optimal policy ($u^*$) is obtained by substituting the optimal value-function ($J^*$) in the Bellman equation. Alternately $u^*$ is also obtained by learning the optimal state-action value function $Q^*$ known as the $Q$ value-function. However, it is difficult to compute the exact values of $J^*$ or $Q^*$ for MDPs with large number of states. Approximate Dynamic Programming (ADP) methods address this difficulty by computing lower dimensional approximations of $J^*$/$Q^*$. Most ADP methods employ linear function approximation (LFA), i.e., the approximate solution lies in a subspace spanned by a family of pre-selected basis functions. The approximation is obtain via a linear least squares projection of higher dimensional quantities and the $L_2$ norm plays an important role in convergence and error analysis. In this paper, we discuss ADP methods for MDPs based on LFAs in $(\min,+)$ algebra. Here the approximate solution is a $(\min,+)$ linear combination of a set of basis functions whose span constitutes a subsemimodule. Approximation is obtained via a projection operator onto the subsemimodule which is different from linear least squares projection used in ADP methods based on conventional LFAs. MDPs are not $(\min,+)$ linear systems, nevertheless, we show that the monotonicity property of the projection operator helps us to establish the convergence of our ADP schemes. We also discuss future directions in ADP methods for MDPs based on the $(\min,+)$ LFAs.

We learn sensor trees from training data to minimize sensor acquisition costs during test time. Our system adaptively selects sensors at each stage if necessary to make a confident classification. We pose the problem as empirical risk minimization over the choice of trees and node decision rules. We decompose the problem, which is known to be intractable, into combinatorial (tree structures) and continuous parts (node decision rules) and propose to solve them separately. Using training data we greedily solve for the combinatorial tree structures and for the continuous part, which is a non-convex multilinear objective function, we derive convex surrogate loss functions that are piecewise linear. The resulting problem can be cast as a linear program and has the advantage of guaranteed convergence, global optimality, repeatability and computational efficiency. We show that our proposed approach outperforms the state-of-art on a number of benchmark datasets.

Linear programming is now included in algorithm undergraduate and postgraduate courses for computer science majors. We give a self-contained treatment of an interior-point method which is particularly tailored to the typical mathematical background of CS students. In particular, only limited knowledge of linear algebra and calculus is assumed.

Graph edit distance (GED) is a powerful and flexible graph matching paradigm that can be used to address different tasks in structural pattern recognition, machine learning, and data mining. In this paper, some new binary linear programming formulations for computing the exact GED between two graphs are proposed. A major strength of the formulations lies in their genericity since the GED can be computed between directed or undirected fully attributed graphs (i.e. with attributes on both vertices and edges). Moreover, a relaxation of the domain constraints in the formulations provides efficient lower bound approximations of the GED. A complete experimental study comparing the proposed formulations with 4 state-of-the-art algorithms for exact and approximate graph edit distances is provided. By considering both the quality of the proposed solution and the efficiency of the algorithms as performance criteria, the results show that none of the compared methods dominates the others in the Pareto sense. As a consequence, faced to a given real-world problem, a trade-off between quality and efficiency has to be chosen w.r.t. the application constraints. In this context, this paper provides a guide that can be used to choose the appropriate method.

F2 Decison Making 2- Limiting Factor And Linear Programming

In previous work, we demonstrated how decoding of a non-binary linear code could be formulated as a linear-programming problem. In this paper, we study different polytopes for use with linear-programming decoding, and show that for many classes of codes these polytopes yield a complexity advantage for decoding. These representations lead to polynomial-time decoders for a wide variety of classical non-binary linear codes.

Theorem 38 and Corollary 39 are in error. The modeling needs 9-dimensional z-variables instead of the 8-dimensional variables defined in notations 24.1. Examples of the correct model (with 9-index variables) are: (1) Diaby, M., "Linear Programming Formulation of the Set Partitioning Problem," International Journal of Operational Research 8:4 (August 2010) pp. 399-427; (2) Diaby, M., "Linear Programming Formulation of the Vertex Coloring Problem," International Journal of Mathematics in Operational Research 2:3 (May 2010) pp. 259-289; (3) Diaby, M., "The Traveling Salesman Problem: A Linear Programming Formulation," WSEAS Transactions on Mathematics, 6:6 (June 2007) pp. 745-754.

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.

We consider the problem of finding (possibly non connected) discrete surfaces spanning a finite set of discrete boundary curves in the three-dimensional space and minimizing (globally) a discrete energy involving mean curvature. Although we consider a fairly general class of energies, our main focus is on the Willmore energy, i.e. the total squared mean curvature Our purpose is to address the delicate task of approximating global minimizers of the energy under boundary constraints. The main contribution of this work is to translate the nonlinear boundary value problem into an integer linear program, using a natural formulation involving pairs of elementary triangles chosen in a pre-specified dictionary and allowing self-intersection. Our work focuses essentially on the connection between the integer linear program and its relaxation. We prove that: - One cannot guarantee the total unimodularity of the constraint matrix, which is a sufficient condition for the global solution of the relaxed linear program to be always integral, and therefore to be a solution of the integer program as well; - Furthermore, there are actually experimental evidences that, in some cases, solving the relaxed problem yields a fractional solution. Due to the very specific structure of the constraint matrix here, we strongly believe that it should be possible in the future to design ad-hoc integer solvers that yield high-definition approximations to solutions of several boundary value problems involving mean curvature, in particular the Willmore boundary value problem.

Index Coding has received considerable attention recently motivated in part by real-world applications and in part by its connection to Network Coding. The basic setting of Index Coding encodes the problem input as an undirected graph and the fundamental parameter is the broadcast rate $\beta$, the average communication cost per bit for sufficiently long messages (i.e. the non-linear vector capacity). Recent nontrivial bounds on $\beta$ were derived from the study of other Index Coding capacities (e.g. the scalar capacity $\beta_1$) by Bar-Yossef et al (2006), Lubetzky and Stav (2007) and Alon et al (2008). However, these indirect bounds shed little light on the behavior of $\beta$: there was no known polynomial-time algorithm for approximating $\beta$ in a general network to within a nontrivial (i.e. $o(n)$) factor, and the exact value of $\beta$ remained unknown for any graph where Index Coding is nontrivial. Our main contribution is a direct information-theoretic analysis of the broadcast rate $\beta$ using linear programs, in contrast to previous approaches that compared $\beta$ with graph-theoretic parameters. This allows us to resolve the aforementioned two open questions. We provide a polynomial-time algorithm with a nontrivial approximation ratio for computing $\beta$ in a general network along with a polynomial-time decision procedure for recognizing instances with $\beta=2$. In addition, we pinpoint $\beta$ precisely for various classes of graphs (e.g. for various Cayley graphs of cyclic groups) thereby simultaneously improving the previously known upper and lower bounds for these graphs. Via this approach we construct graphs where the difference between $\beta$ and its trivial lower bound is linear in the number of vertices and ones where $\beta$ is uniformly bounded while its upper bound derived from the naive encoding scheme is polynomially worse.

Hyperbolic Programming (HP) --minimizing a linear functional over an affine subspace of a finite-dimensional real vector space intersected with the so-called hyperbolicity cone-- is a class of convex optimization problems that contains well-known Linear Programming (LP). In particular, for any LP one can readily provide a sequence of HP relaxations. Based on these hyperbolic relaxations, a new Shrink-Wrapping approach to solve LP has been proposed by Renegar. The resulting Shrink-Wrapping trajectories, in a sense, generalize the notion of central path in interior-point methods. We study the geometry of Shrink-Wrapping trajectories for Linear Programming. In particular, we analyze the geometry of these trajectories in the proximity of the so-called central line, and contrast the behavior of these trajectories with that of the central path for some pathological LP instances. In addition, we provide an elementary proof of convexity of hyperbolicity cones over reals.

Bell System Technical Journal, 38: 6 November 1959 pp 1485-1512. Error-Correcting Codes--A Linear Programming Approach (McCluskey, E.J., Jr.)

Bell System Technical Journal, 38: 6 November 1959 pp 1485-1512. Error-Correcting Codes--A Linear Programming Approach (McCluskey, E.J., Jr.)

Bell System Technical Journal, 38: 6 November 1959 pp 1485-1512. Error-Correcting Codes--A Linear Programming Approach (McCluskey, E.J., Jr.)

Software-defined networks have been proposed as a viable solution to decrease the power consumption of the networking component in data center networks. Still the question remains on which scheduling algorithms are most suited to achieve this goal. We propose 4 different linear programming approaches that schedule requested traffic flows on SDN switches according to different objectives. Depending on pre-defined software quality requirements such as delay and performance, a single variation or a combination of variations can be selected to optimize the power saving and the performance metrics. Our simulation results demonstrate that all our algorithm variations outperform the shortest path scheduling algorithm, our baseline on power savings, less or more strongly depending on the power model chosen. We show that in FatTree networks, where switches can save up to 60% of power in sleeping mode, we can achieve 15% minimum improvement assuming a one-to-one traffic scenario. Two of our algorithm variations privilege performance over power saving and still provide around 45% of the maximum achievable savings.

40 leaves

40 leaves

As part of the management planning and control function, the U.S. Coast Guard's Pacific Area Commander schedules the operational missions for all High Endurance Cutters in the Pacific Area. To provide a powerful management tool to assist this scheduling process, an analytic model for this large scale problem has been developed and implemented. It contains mission requirements, restricted sequencing of missions, ships' physical limitations and crews' morale-related considerations. The modeling approach is based on the Geoffrion-Graves model for parallel production lines with significant changeover costs. The implementation solves a large (860 row) Koopmans-Beckmann fixed charge Quadratic Assignment model using a new method with an advanced, feasible starting solution provided by an imbedded network (with 1,720 nodes and 739,600 arcs). Many linear programming problems (200 row, 450 variable) are then solved with a linear programming subroutine of advanced design. The resulting model and these implementation techniques produce excellent quality working schedules with very reasonable execution time and memory requirements. Alternative solutions are easily generated

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

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

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

Outsourcing of scientific computations is attracting increasing attention since it enables the customers with limited computing resource and storage devices to outsource the sophisticated computation workloads into powerful service providers. However, it also comes up with some security and privacy concerns and challenges, such as the input and output privacy of the customers, and cheating behaviors of the cloud. Motivated by these issues, this paper focused on privacy-preserving Linear Fractional Programming (LFP) as a typical and practically relevant case for verifiable secure multiparty computation. We will investigate the secure and verifiable schema with correctness guarantees, by using normal multiparty techniques to compute the result of a computation and then using verifiable techniques only to verify that this result was correct.

We propose new estimates for the frontier of a set of points. They are defined as kernel estimates covering all the points and whose associated support is of smallest surface. The estimates are written as linear combinatio- ns of kernel functions applied to the points of the sample. The coefficients of the linear combination are then computed by solving a linear programming problem. In the general case, the solution of the optimizat- ion problem is sparse, that is, only a few coefficients are non zero. The corresponding points play the role of support vectors in the statistical learning theory. The L_1 error between the estimated and the true frontiers is shown to be almost surely converging to zero, and the rate of convergence is provided. The behaviour of the estimates on finite sample situations is illustrated on some simulations.

We propose new estimates for the frontier of a set of points. They are defined as kernel estimates covering all the points and whose associated support is of smallest surface. The estimates are written as linear combinatio- ns of kernel functions applied to the points of the sample. The coefficients of the linear combination are then computed by solving a linear programming problem. In the general case, the solution of the optimizat- ion problem is sparse, that is, only a few coefficients are non zero. The corresponding points play the role of support vectors in the statistical learning theory. The L_1 error between the estimated and the true frontiers is shown to be almost surely converging to zero, and the rate of convergence is provided. The behaviour of the estimates on finite sample situations is illustrated on some simulations.

We propose new estimates for the frontier of a set of points. They are defined as kernel estimates covering all the points and whose associated support is of smallest surface. The estimates are written as linear combinatio- ns of kernel functions applied to the points of the sample. The coefficients of the linear combination are then computed by solving a linear programming problem. In the general case, the solution of the optimizat- ion problem is sparse, that is, only a few coefficients are non zero. The corresponding points play the role of support vectors in the statistical learning theory. The L_1 error between the estimated and the true frontiers is shown to be almost surely converging to zero, and the rate of convergence is provided. The behaviour of the estimates on finite sample situations is illustrated on some simulations.

A crucial aspect of the Box Method for linear programming is the finding of a minimum-weight basis corresponding to a given interior feasible point. This subproblem leads to the formation of the Box Problem, a special linear program having a closed form solution which provides the search direction at the current iteration. Finding a minimum-weight basis is a matroidal(or, combinatorial) optimization problem that can be handled by a greedy algorithm. This paper suggests a way of efficiently solving the minimum-weight basis problem in cases where the (primal, standard form) linear program contains explicitly bounded variables. It is shown that the main part of the task requires almost no more computational effort or storage space than does a problem of the same size without upper bounded variables. While this result is believed to be valuable in its own right, there is additional benefit to be gained in applications where the finding of a minimum-weight basis (for a linear program without explicit upper bounds on its variables) is done by a special greedy algorithm. Such is the case with minimum-cost network flow problems which will be discussed in Part III of this series.

In this report the author presents a heuristic for constructing surrogate constraints to be used for the solution of the linear zero-one integer problem. Using the heuristic the author was able to build surrogate constraints with strength comparable to the dual multiplier surrogate in one-tenth the time. (Author)

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.

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.

Experience with solving a 12,753,313 variable linear program is described. This problem is the linear programming relaxation of a set partitioning problem arising from an airline crew scheduling application. A scheme is described that requires successive solutions of small subproblems, yielding a procedure that has little growth in solution time in terms of the number of variables. Experience using the simplex method as implemented in CPLEX, an interior point method as implemented in OB1, and a hybrid interior point/simplex approach is reported. The resulting procedure illustrates the power of an interior point/simplex combination for solving very large-scale linear programs.

The optimal timing of air-to-ground tasks is undertaken. Specifically, a scenario where multiple air vehicles are required to prosecute geographically dispersed targets is considered. The vehicles must perform multiple tasks on each target. The targets must be found, classified, attacked, and verified as destroyed. The optimal performance of these tasks requires cooperation amongst the vehicles such that critical timing constraints are satisfied. In this paper, an optimal task assignment and timing algorithm is developed, using a mixed integer linear program, or MILP, formulation. MILP can be used to assign all tasks to the vehicles in an optimal manner, including variable arrival times, for groups of air vehicles with coupled tasks involving timing and task order constraints. When the air vehicles have sufficient endurance, the existence of a solution is guaranteed.

Signal to Noise Ratio (SNR) is an important index for wireless communications. In CDMA systems, spreading sequences are utilized. This series of papers show the method to derive spreading sequences as the solutions of non-linear programming: maximize SNR. In this paper, we derive the optimization problems with the expression SNR derived in Part I and the necessary conditions for the global solutions. We numerically solve the problems and evaluate the solutions with two factors, mean-square correlations and maximum mean-square correlations.