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Linear and Nonlinear Programming by David G.Luenberger and Yinyu Ye

This paper discusses the calculation of sensitivities. or derivatives, for optimization problems involving systems governed by differential equations and other state relations. The subject is examined from the point of view of nonlinear programming, beginning with the analytical structure of the first and second derivatives associated with such problems and the relation of these derivatives to implicit differentiation and equality constrained optimization. We also outline an error analysis of the analytical formulae and compare the results with similar results for finite-difference estimates of derivatives. We then attend to an investigation of the nature of the adjoint method and the adjoint equations and their relation to directions of steepest descent. We illustrate the points discussed with an optimization problem in which the variables are the coefficients in a differential operator.

The fuel optimal control problem arising in coplanar orbital transfer employing aeroassisted technology is addressed. The mission involves the transfer from high energy orbit (HEO) to low energy orbit (LEO) without plane change. The basic approach here is to employ a combination of propulsive maneuvers in space and aerodynamic maneuvers in the atmosphere. The basic sequence of events for the coplanar aeroassisted HEO to LEO orbit transfer consists of three phases. In the first phase, the transfer begins with a deorbit impulse at HEO which injects the vehicle into a elliptic transfer orbit with perigee inside the atmosphere. In the second phase, the vehicle is optimally controlled by lift and drag modulation to satisfy heating constraints and to exit the atmosphere with the desired flight path angle and velocity so that the apogee of the exit orbit is the altitude of the desired LEO. Finally, the second impulse is required to circularize the orbit at LEO. The performance index is maximum final mass. Simulation results show that the coplanar aerocapture is quite different from the case where orbital plane changes are made inside the atmosphere. In the latter case, the vehicle has to penetrate deeper into the atmosphere to perform the desired orbital plane change. For the coplanar case, the vehicle needs only to penetrate the atmosphere deep enough to reduce the exit velocity so the vehicle can be captured at the desired LEO. The peak heating rates are lower and the entry corridor is wider. From the thermal protection point of view, the coplanar transfer may be desirable. Parametric studies also show the maximum peak heating rates and the entry corridor width are functions of maximum lift coefficient. The problem is solved using a direct optimization technique which uses piecewise polynomial representation for the states and controls and collocation to represent the differential equations. This converts the optimal control problem into a nonlinear programming problem which is solved numerically by using a modified version of NPSOL. Solutions were obtained for the described problem for cases with and without heating constraints. The method appears to be more robust than other optimization methods. In addition, the method can handle complex dynamical constraints.

This report details the development of a reliability based multi-objective design tool for solving structural optimization problems. Based on two different optimization techniques, namely sequential unconstrained minimization and nonlinear goal programming, the developed design method has the capability to take into account the effects of variability on the proposed design through a user specified reliability design criterion. In its sequential unconstrained minimization mode, the developed design tool uses a composite objective function, in conjunction with weight ordered design objectives, in order to take into account conflicting and multiple design criteria. Multiple design criteria of interest including structural weight, load induced stress and deflection, and mechanical reliability. The nonlinear goal programming mode, on the other hand, provides for a design method that eliminates the difficulty of having to define an objective function and constraints, while at the same time has the capability of handling rank ordered design objectives or goals. For simulation purposes the design of a pressure vessel cover plate was undertaken as a test bed for the newly developed design tool. The formulation of this structural optimization problem into sequential unconstrained minimization and goal programming form is presented. The resulting optimization problem was solved using: (i) the linear extended interior penalty function method algorithm; and (ii) Powell's conjugate directions method. Both single and multi-objective numerical test cases are included demonstrating the design tool's capabilities as it applies to this design problem.

This paper investigates the relation between sequential convex programming (SCP) as, e.g., defined in [24] and DC (difference of two convex functions) programming. We first present an SCP algorithm for solving nonlinear optimization problems with DC constraints and prove its convergence. Then we combine the proposed algorithm with a relaxation technique to handle inconsistent linearizations. Numerical tests are performed to investigate the behaviour of the class of algorithms.

This paper studies the robust optimal control design for uncertain nonlinear systems from a perspective of robust adaptive dynamic programming (robust-ADP). The objective is to fill up a gap in the past literature of ADP where dynamic uncertainties or unmodeled dynamics are not addressed. A key strategy is to integrate tools from modern nonlinear control theory, such as the robust redesign and the backstepping techniques as well as the nonlinear small-gain theorem, with the theory of ADP. The proposed robust-ADP methodology can be viewed as a natural extension of ADP to uncertain nonlinear systems. A practical learning algorithm is developed in this paper, and has been applied to a sensorimotor control problem.

A new computational method is presented for calculating the phases required for minimized weight perturbation, phase-only null synthesis in linear array antenna patterns. The method uses nonlinear programming algorithms to determine the coefficients of a beam space representation of the optimal phase perturbations. The derivatives of the objective and null constraint functions with respect to the beam coefficients, required by the nonlinear programming algorithms, are calculated from analytic expressions. The beam space representation allows the number of unknowns to be reduced from half the number of array elements to the number of imposed null locations, and results in a significant reduction in computation time from that required to calculate the phase perturbations directly.

The problem of calculating the minimum phase-only weight perturbations required to impose nulls in the pattern of a linear antenna array is nonlinear and cannot be solved analytically. The problem is, however, an example of nonlinear programming problem and can be solved numerically. This report describes the performance of two nonlinear programming computer codes, LPNLP and VMCON, on the minimized weight perturbation phase-only null synthesis problem. Both codes are effective in general, although convergence problems may be encountered if nulls are required to be imposed at closely spaced locations in high sidelobe regions of the pattern.

Any real-valued nonlinear function in 0-1 variables can be rewritten as a multilinear function. We discuss classes of lower and upper bounding linear expressions for multilinear functions in 0-1 variables. For any multilinear inequality in 0-1 variables, we define an equivalent family of linear inequalities. This family contains the well known system of generalized covering inequalities, as well as other linear equivalents of the multilinear inequality that are more compact, i.e., of smaller cardinality. In a companion paper, we discuss dominance relations between various linear equivalents of a multilinear inequality, and describe a class of algorithms for multilinear 0-1 programming based on these results. (Author)

We study the generalization of split and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference of two convex sets with specific geometric structures. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split and intersection cuts for several classes of sets. In particular, we give simple formulas for split cuts for essentially all convex sets described by a single quadratic inequality and for more general intersection cuts for a wide variety of convex quadratic sets.

We study the generalization of split and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference of two convex sets with specific geometric structures. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split and intersection cuts for several classes of sets. In particular, we give simple formulas for split cuts for essentially all convex sets described by a single quadratic inequality and for more general intersection cuts for a wide variety of convex quadratic sets.

Bibliography: p. 238

Nonlinear programming methods are used to solve saddle-point problems subject to inequality constraints on the variables; in particular, the type of saddle-point problem arising in pursuit-evasion differential games is considered. The methods investigated fall into two groups: solution of the nonlinear simultaneous equations obtained from the Kuhn-Tucker conditions, and solution of a sequence of constrained optimization problems by the gradient projection algorithm. These methods are applicable to any real-valued function f(x,y) which is convex in x, concave in y, and has continuous and bounded second partial derivatives. Several examples are given which illustrate the characteristics of the numerical procedures .

Nonlinear programming methods are used to solve saddle-point problems subject to inequality constraints on the variables; in particular, the type of saddle-point problem arising in pursuit-evasion differential games is considered. The methods investigated fall into two groups: solution of the nonlinear simultaneous equations obtained from the Kuhn-Tucker conditions, and solution of a sequence of constrained optimization problems by the gradient projection algorithm. These methods are applicable to any real-valued function f(x,y) which is convex in x, concave in y, and has continuous and bounded second partial derivatives. Several examples are given which illustrate the characteristics of the numerical procedures .

One important method for finding the optimal solution for linear and nonlinear models is the graphical method, which is used if the linear or nonlinear mathematical model contains one, two, or three variables. Models that contain only two variables are models for which the optimal solution has been obtained graphically, whether these models are linear or non-linear in references that rely on classical logic. In this research, we present a study through which we present the graphical method for solving nonlinear neutrosophical models in the following case: (1) The objective function is a nonlinear function, and the constraints are linear functions. (2) The objective function is a linear function and the constraints are non-linear. (3) The objective function is a non-linear function and the constraints are non-linear. In the three cases, the optimal solution is the vector that satisfies all the constraints and at which the function reaches a maximum or minimum value depending on the nature of the subject under study (note that it does not have to be unique).

Book Source: Digital Library of India Item 2015.192686 dc.contributor.author: M.l.malik dc.date.accessioned: 2015-07-08T01:32:06Z dc.date.available: 2015-07-08T01:32:06Z dc.date.digitalpublicationdate: 2005-08-20 dc.identifier.barcode: 1990010091087 dc.identifier.origpath: /rawdataupload/upload/0091/087 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/192686 dc.description.scannerno: 12 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 112 dc.format.mimetype: application/pdf dc.language.iso: English dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Electrical Engineering dc.title: Linear And Nonlinear Programming On Analog Computer

Book Source: Digital Library of India Item 2015.192686 dc.contributor.author: M.l.malik dc.date.accessioned: 2015-07-08T01:32:06Z dc.date.available: 2015-07-08T01:32:06Z dc.date.digitalpublicationdate: 2005-08-20 dc.identifier.barcode: 1990010091087 dc.identifier.origpath: /rawdataupload/upload/0091/087 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/192686 dc.description.scannerno: 12 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 112 dc.format.mimetype: application/pdf dc.language.iso: English dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Electrical Engineering dc.title: Linear And Nonlinear Programming On Analog Computer

Book Source: Digital Library of India Item 2015.192686 dc.contributor.author: M.l.malik dc.date.accessioned: 2015-07-08T01:32:06Z dc.date.available: 2015-07-08T01:32:06Z dc.date.digitalpublicationdate: 2005-08-20 dc.identifier.barcode: 1990010091087 dc.identifier.origpath: /rawdataupload/upload/0091/087 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/192686 dc.description.scannerno: 12 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 112 dc.format.mimetype: application/pdf dc.language.iso: English dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Electrical Engineering dc.title: Linear And Nonlinear Programming On Analog Computer

This volume was digitized and made accessible online due to deterioration of the original print copy.

This paper presents an interactive fuzzy goal programming (FGP) approach for solving multiobjective nonlinear programming problems (MONLPP) with interval type 2 fuzzy numbers (IT2 FNs). The cost and time of the objective functions, the resources, and the requirements of each kind of resources are taken to be trapezoidal IT2 FNs. Here, the considered problem is first transformed into an equivalent crisp MONLPP, and then the transformed MONLPP is converted into an equivalent Multiobjective Linear Programming Problem (MOLPP). By using a procedure based on Taylor series, this problem is reduced into a single objective linear programming problem (LPP) which can be easily solved by Maple 18.02 optimization toolbox. Finally, the proposed solution procedure is illustrated by two numerical examples.

Several algorithms for solving problems in linear, quadratic, and nonlinear programming, network flows, and facilities location were developed and analyzed. Results include: (1) The analysis of the computational complexity of the problem of determining an optimally sparse representation of the null space of a matrix, and the development of worst-case bounds for the shadow-vertex simplex algorithm and several heuristics for distance constrained discrete facility location problems and conditional covering problems; (2) The development of efficient algorithms for dense and sparse assignment problems, strictly convex quadratic programming problems, and a nonlinear programming problem that arises when maximizing a correlation coefficient subject to linear constraints; and (3) The application of iterative methods to large sparse equality-constraind quadratic programs and the development of multiple constraint deletion strategies for active-set algorithms for linearly consrained nonlinear programming problems. (Author)

The investigators associated with the research reported herein have been supported by ARO almost continuously over the past 20 years, from the inception of their early development of penalty function and branch and bound methodology. This report covers results developed from 1 January 1974 to 31 May 1981 under two ARO contracts and one grant, results dealing mainly with nonconvex programming, second order algorithms and sensitivity and stability methodology. Technical details are omitted, the main intent being the provision of a concise chronicle of the major accomplishments.

A nonlinear 0-1 program can be restated as a multilinear 0-1 program, which in turn is known to be equivalent to a linear 0-1 program with generalized covering (g.c.) inequalities. In a companion paper 6 we have defined a family of linear inequalities that contains more compact (smaller cardinality) linearizations of a multilinear 0-1 program than the one based on the g.c. inequalities. In this paper we analyze the dominance relations between inequalities of the above family. In particular, we give a criterion that can be checked in linear time, for deciding whether a g.c. inequality can be strengthened by extending the cover from which it was derived. We then describe a class of algorithms based on these results and discuss our computational experience. We conclude that the g.c. inequalities can be strengthened most of the time an extent that increases with problem density. In particular, the algorithm using the strengthening procedure outperforms the one using only g.c. inequalities whenever the number of nonlinear terms per constraint exceeds about 12-15, and the difference in their performance grows with the number of such terms. (Author)

This report forms the user's guide of SOL/NPSOL, a set of Fortran subroutines designed to minimize an arbitrary smooth function subject to constraints, which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints (NPSOL may also be used for uncontrained, bound-constrained and linearly constrained optimization.) The user must provide subroutines that define the objective and constraint functions and their gradients. All matrices are treated as dense, and hence NPSOL is not intended for large sparse problems. NPSOL uses a sequential quadratic programming (SQP) algorithm, in which the search direction is the solution of a quadratic programming (QP) subproblem. The algorithm treats bounds, linear constraints and nonlinear constraints separately. The Hessian of each QP subproblem is a positive-definite quasi-Newton approximation to the Hessian of an augmented Lagrangian function. The steplength at each iteration is required to produce a sufficient decrease in an augmented Lagrangian merit function. Each QP subproblem is solved using a quadratic programming package with several features that improve the efficiency of an SQP algorithm.

The fuel optimal control problem arising in the non-planar orbital transfer employing aeroassisted technology is addressed. The mission involves the transfer from high energy orbit (HEO) to low energy orbit (LEO) with orbital plane change. The basic strategy here is to employ a combination of propulsive maneuvers in space and aerodynamic maneuvers in the atmosphere. The basic sequence of events for the aeroassisted HEO to LEO transfer consists of three phases. In the first phase, the orbital transfer begins with a deorbit impulse at HEO which injects the vehicle into an elliptic transfer orbit with perigee inside the atmosphere. In the second phase, the vehicle is optimally controlled by lift and bank angle modulations to perform the desired orbital plane change and to satisfy heating constraints. Because of the energy loss during the turn, an impulse is required to initiate the third phase to boost the vehicle back to the desired LEO orbital altitude. The third impulse is then used to circularize the orbit at LEO. The problem is solved by a direct optimization technique which uses piecewise polynomial representation for the state and control variables and collocation to satisfy the differential equations. This technique converts the optimal control problem into a nonlinear programming problem which is solved numerically. Solutions were obtained for cases with and without heat constraints and for cases of different orbital inclination changes. The method appears to be more powerful and robust than other optimization methods. In addition, the method can handle complex dynamical constraints.

Certain new approaches to linear programming have recently received considerable publicity because of the promise of substantial improvements in efficiency compared to the simplex method. This note briefly discusses several research directions in methods for solving linear programs using nonlinear problem transformations. In particular, we describe application of a barrier transformation to the dual, and the development of sparse least-squares methods based on the LU factorization of the least-squares matrix of its transpose. (Author)

The project explored the theoretical properties and computational performance of algorithms for solving constrained optimization problems (linear and nonlinear programs). Particular emphasis was placed on algorithms for solving large problems. The practical applications of optimization are innumerable. For example, mathematical models of the economy (to analyze the optimal use of natural resources) are typically large linear or nonlinear programs. Areas in which we have been actively involved include optimal generation and transmission of electricity, optimization of aircraft and spacecraft trajectories, optimal structural design, and financial modeling such as portfolio optimization. Progress on solution algorithms and software for such applications is ultimately reflected in improved techniques in many other areas of science and industry.

The project explored the theoretical properties and computational performance of algorithms for solving constrained optimization problems (linear and nonlinear programs). Particular emphasis was placed on algorithms for solving large problems. The practical applications of optimization are innumerable. For example, mathematical models of the economy (to analyze the optimal use of natural resources) are typically large linear or nonlinear programs. Areas in which we have been actively involved include optimal generation and transmission of electricity, optimization of aircraft and spacecraft trajectories, optimal structural design, and financial modeling such as portfolio optimization. Progress on solution algorithms and software for such applications is ultimately reflected in improved techniques in many other areas of science and industry.

The project explored the theoretical properties and computational performance of algorithms for solving constrained optimization problems (linear and nonlinear programs). Particular emphasis was placed on algorithms for solving large problems. The practical applications of optimization are innumerable. For example, mathematical models of the economy (to analyze the optimal use of natural resources) are typically large linear or nonlinear programs. Areas in which we have been actively involved include optimal generation and transmission of electricity, optimization of aircraft and spacecraft trajectories, optimal structural design, and financial modeling such as portfolio optimization. Progress on solution algorithms and software for such applications is ultimately reflected in improved techniques in many other areas of science and industry.

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A new method is given for preventing the simplex method from cycling. Key features are that a positive step is taken at every iteration, and nonbasic variables are allowed to be slightly infeasible. There is no additional work per iteration. Computational results are given for the first 53 test problems in netlib, indicating reliable performance in all cases. The method may be applied to active-set methods for solving nonlinear programs with linear constraints. Keywords: EXPAND procedure, EXPAND(Expanding Tolerance Anti-Degeneracy), Optimization.

The United States Air Force uses a nonlinear programming model to assess the utilization of weapons and sorties needed to achieve a maximum value of destroyed targets in a multi-period, Theater-Level conflict. The current model is modified by constraining the consumption of weapons. Alternate objective functions are introduced. Their meaning and influence on the optimization is compared. An increase in the worth of destroyed targets is gained if the model can more flexibly utilize weapons than is currently the case. The optimization can be further improved if all time periods are considered simultaneously while assigning sorties to targets, rather than the current myopic approach. Keywords: Theses; Global optimization, Sortie allocation.

We consider control design for positive compartmental systems in which each compartment's outflow rate is described by a concave function of the amount of material in the compartment.We address the problem of determining the routing of material between compartments to satisfy time-varying state constraints while ensuring that material reaches its intended destination over a finite time horizon. We give sufficient conditions for the existence of a time-varying state-dependent routing strategy which ensures that the closed-loop system satisfies basic network properties of positivity, conservation and interconnection while ensuring that capacity constraints are satisfied, when possible, or adjusted if a solution cannot be found. These conditions are formulated as a linear programming problem. Instances of this linear programming problem can be solved iteratively to generate a solution to the finite horizon routing problem. Results are given for the application of this control design method to an example problem. Key words: linear programming; control of networks; positive systems; controller constraints and structure.

An algorithm is presented for solution of dynamic optimization problems which are nonlinear in the state variables and linear in the control variables. It is shown that the optimal control is bang-bang. A nominal bang-bang solution is found which satisfies the system equations and constraints, and influence functions are generated which check the optimality of the solution. Nonlinear optimization (gradient search) techniques are used to find the optimal solution. The algorithm is used to find a minimum time acceleration for a turbofan engine.

An algorithm is presented for solution of dynamic optimization problems which are nonlinear in the state variables and linear in the control variables. It is shown that the optimal control is bang-bang. A nominal bang-bang solution is found which satisfies the system equations and constraints, and influence functions are generated which check the optimality of the solution. Nonlinear optimization (gradient search) techniques are used to find the optimal solution. The algorithm is used to find a minimum time acceleration for a turbofan engine.

An algorithm is presented for solution of dynamic optimization problems which are nonlinear in the state variables and linear in the control variables. It is shown that the optimal control is bang-bang. A nominal bang-bang solution is found which satisfies the system equations and constraints, and influence functions are generated which check the optimality of the solution. Nonlinear optimization (gradient search) techniques are used to find the optimal solution. The algorithm is used to find a minimum time acceleration for a turbofan engine.

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Contraction analysis is a stability theory for nonlinear systems where stability is defined incrementally between two arbitrary trajectories. It provides an alternative framework in which to study uncertain interconnections or systems with external inputs, where it offers several significant advantages when compared with traditional Lyapunov analysis. Contraction-based methods are particularly useful for analyzing systems with uncertain parameters and for proving synchronization properties of nonlinear oscillators. Existence of a contraction metric for a given system is a necessary and sufficient condition for global exponential convergence of system trajectories. For systems with polynomial or rational dynamics, the search for contraction metrics can be made fully algorithmic through the use of convex optimization and sum of squares (SOS) programming. The search process is made computationally tractable by relaxing matrix definiteness constraints, whose feasibility indicate existence of a contraction metric, into SOS constraints on polynomial matrices. We illustrate the results through examples from the literature, emphasizing the advantages and contrasting the differences between the contraction approach and traditional Lyapunov techniques.

This volume was digitized and made accessible online due to deterioration of the original print copy.

Bibliography: l. 79-81

Minimum time accelerations of aircraft turbofan engines are presented. The calculation of these accelerations was made by using a piecewise linear engine model, and an algorithm based on nonlinear programming. Use of this model and algorithm allows such trajectories to be readily calculated on a digital computer with a minimal expenditure of computer time.

Minimum time accelerations of aircraft turbofan engines are presented. The calculation of these accelerations was made by using a piecewise linear engine model, and an algorithm based on nonlinear programming. Use of this model and algorithm allows such trajectories to be readily calculated on a digital computer with a minimal expenditure of computer time.

Minimum time accelerations of aircraft turbofan engines are presented. The calculation of these accelerations was made by using a piecewise linear engine model, and an algorithm based on nonlinear programming. Use of this model and algorithm allows such trajectories to be readily calculated on a digital computer with a minimal expenditure of computer time.

Packing and vehicle routing problems play an important role in the area of supply chain management. In this paper, we introduce a non-linear knapsack problem that occurs when packing items along a fixed route and taking into account travel time. We investigate constrained and unconstrained versions of the problem and show that both are NP-hard. In order to solve the problems, we provide a pre-processing scheme as well as exact and approximate mixed integer programming (MIP) solutions. Our experimental results show the effectiveness of the MIP solutions and in particular point out that the approximate MIP approach often leads to near optimal results within far less computation time than the exact approach.

Packing and vehicle routing problems play an important role in the area of supply chain management. In this paper, we introduce a non-linear knapsack problem that occurs when packing items along a fixed route and taking into account travel time. We investigate constrained and unconstrained versions of the problem and show that both are NP-hard. In order to solve the problems, we provide a pre-processing scheme as well as exact and approximate mixed integer programming (MIP) solutions. Our experimental results show the effectiveness of the MIP solutions and in particular point out that the approximate MIP approach often leads to near optimal results within far less computation time than the exact approach.

[REVISED VERSION] The aim of this paper is to state a sharp version of the K\"onig supremum theorem, an equivalent reformulation of the Hahn--Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush-Kuhn-Tucker and Fritz John type, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.

[REVISED VERSION] The aim of this paper is to state a sharp version of the K\"onig supremum theorem, an equivalent reformulation of the Hahn--Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush-Kuhn-Tucker and Fritz John type, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.