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GAMS/CONOPT has a fast method for finding a first feasible solution that is particularly well suited for models with few degrees of freedom. If you have a model with roughly the same number of constraints as variable you should try CONOPT. CONOPT can also be used to solve square systems of equations without an objective function corresponding to the GAMS model class CNS - Constrained Nonlinear System.

GAMS/CONOPT can use second derivatives. If the number of variables is much larger than the number of constraints CONOPT will use second derivatives and overall progress can be considerably faster than for MINOS or SNOPT. IPOPT and KNITRO will also use second derivatives, but the method is very different and it is not possible to predict which solver will be better.

GAMS/CONOPT has been designed for large and sparse models. This means that both the number of variables and equations can be large. Indeed, NLP models with over 100,000 equations and variables have been solved successfully, and CNS models with over 1,000,000 equations and variables have also been solved. The components used to build CONOPT have been selected under the assumptions that the model is sparse, i.e. that most functions only depend on a small number of variables. CONOPT can also be used for denser models, but the performance will suffer significantly.

The remaining iterations are characterized by the value of \"Phase\" in column 2. The model is infeasible during Phase 0, 1, and 2 and the Sum of Infeasibilities in column 4 (labeled \"Infeasibility\") is being minimized; the model is feasible during Phase 3 and 4 and the actual objective function, also shown in column 4 (now labeled \"Objective\"), is minimized or maximized. Phase 0 iterations are Newton-like iterations. They are very cheap so you should not be concerned if there are many of these Phase 0 iterations. During Phase 1 and 3 the model behaves almost linearly and CONOPT applies special linear iterations that take advantage of the linearity. These iterations are sometimes augmented with some inner \"Sequential Linear Programming\" (SLP) iterations, indicated by a number of inner SLP iterations in the \"InItr\" column. During Phase 2 and 4 the model behaves more nonlinear and most aspects of the iterations are therefore changed: the line search is more elaborate, and CONOPT needs second order information to improve the convergence. For small and simple models CONOPT will approximate second order information as a byproduct of the line searches. For larger and more complex models CONOPT will use some inner \"Sequential Quadratic Programming\" (SQP) iterations based on exact second derivatives. These SQP iterations are identified by the number of inner SQP iterations in the \"InItr\" column.

GAMS/CONOPT may terminate in a number of ways. This section will show most of the termination messages and explain their meaning. It will also show the Model Status returned to GAMS in .ModelStat, where represents the name of the GAMS model. The Solver Status returned in .SolveStat will be given if it is different from 1 (Normal Completion). We will in all cases first show the message from CONOPT followed by a short explanation. The first 4 messages are used for optimal solutions and CONOPT will return ModelStat = 2 (Locally Optimal), except as noted below:

The number of function evaluation errors has reached the limit defined in GAMS by Option DomLim = xx; or .DomLim = xx; statements or the default limit of 0 function evaluation errors. CONOPT will return with SolveStat = 5 (Evaluation Error Limit) and ModelStat either 6 (Locally Infeasible) or 7 (Feasible Solution). See more details in section Function Evaluation Errors .

It is recommended that you use the WorkFactor option if you must change the amount of memory. The same number will usually work for a whole family of models. If you prefer to use WorkSpace, the GAMS WorkSpace option corresponds to the amount of memory, measured in Mbytes.

Whenever a nonlinear function is called outside its domain of definition, GAMS' function evaluator will intercept the function evaluation error and prevent the system to crash. GAMS will replace the undefined result by some appropriate real number, and it will make sure the error is reported to the modeler as part of the standard solution output in the GAMS listing file. GAMS will also report the error to CONOPT, so CONOPT can try to correct the problem by backtracking to a safe point. Finally, CONOPT will be instructed to stop after DomLim errors.

The value must be written using legal GAMS format, i.e. a real number may contain an optional E exponent, but a number may not contain blanks. The value must have the same type as the option, i.e. real options must be assigned real values, integer options must be assigned integer values, and logical options must be assigned logical values. The logical value representing true are true, t, yes, or 1, and the logical values representing false are false, f, no, or 0.

The initial values used by CONOPT are all coming from GAMS. The initial values used by GAMS are by default the value zero projected on the bounds. I.e. if a variable is free or has a lower bound of zero, then its default initial value is zero. Unfortunately, zero is in many cases a bad initial value for a nonlinear variable. An initial value of zero is especially bad if the variable appears in a product term since the initial derivative becomes zero, and it appears as if the function does not depend on the variable. CONOPT will warn you and ask you to supply better initial values if the number of derivatives equal to zero is larger than 20 percent.

for three reasons. First, because the number of nonlinear derivatives is reduced in number and complexity. Second, because the lower bound on the intermediate result will bound the search away from the singularity at xSum = 0. And third, because the matrix of second derivatives for the last model only depend on xSum while it depends on all x in the first model.

The reformulated model has n additional variables and n additional linear constraints. In return, the original n complex nonlinear constraints have been changed into n simpler nonlinear constraints. And the number of Jacobian elements, that has a direct influence on much of the computational work both in GAMS and in CONOPT, has been reduced from n*(n+1)/2 nonlinear elements to 3*n-1 linear elements and only n nonlinear element. If f is an invertible increasing function you may even rewrite the last constraint as a simple bound:

GAMS' scaling is in most respects hidden for the modeler. The solution values reported back from a solution algorithm, both primal and dual, are always reported in the user's notation. The algorithm's versions of the equations and variables are only reflected in the derivatives in the equation and column listings in the GAMS output if Option LimRow and/or LimCol are positive, and in debugging output from the solution algorithm, generated with Option SysOut = On. In addition, the numbers in the algorithms iteration log will represent the scaled model: the infeasibilities and reduced gradients will correspond to the scaled model, and if the objective variable is scaled, the value of the objective function will be the scaled value.

One example of a smooth expression involving an Abs function is common in water systems modeling. The pressure loss over a pipe, dH, is proportional to the flow, Q, to some power, P. P is usually around +2. The sign of the loss depend on the direction of the flow so dH is positive if Q is positive and negative if Q is negative. Although GAMS has a Sign function, it cannot be used in a model because of its discontinuous nature. Instead, the pressure loss can be modeled with the equation dH =E= const * Q * Abs(Q)**(P-1), where the sign of the Q-term takes care of the sign of dH, and the Abs function guaranties that the real power ** is applied to a non-negative number. Although the expression involves the Abs function, the derivatives are smooth as long as P is greater than 1. The derivative with respect to Q is const * (P-1) * Abs(Q)**(P-1) for Q > 0 and -const * (P-1) * Abs(Q)**(P-1) for Q < 0. The limit for Q going to zero from both right and left is 0, so the derivative is smooth in the critical point Q = 0 and the overall model is therefore smooth.

where x is the vector of optimization variables, lo and up are vectors of lower and upper bounds, some of which may be minus or plus infinity, b is a vector of right hand sides, and f and g are differentiable nonlinear functions that define the model. n will in the following denote the number of variables and m the number of eq