optim_doc: fmincon

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### 7.6 Non-linear scalar optimization

Function File: fmincon (objf, x0)
Function File: fmincon (objf, x0, A, b)
Function File: fmincon (objf, x0, A, b, Aeq, beq)
Function File: fmincon (objf, x0, A, b, Aeq, beq, lb, ub)
Function File: fmincon (objf, x0, A, b, Aeq, beq, lb, ub, nonlcon)
Function File: fmincon (objf, x0, A, b, Aeq, beq, lb, ub, nonlcon, options)
Function File: fmincon (problem)
Function File: [x, fval, cvg, outp] = fmincon (…)

  Compatibility frontend for nonlinear minimization of a scalar objective function.

  This function is for Matlab compatibility and provides a subset of the functionality of `nonlin_min`.

  objf: objective function. It gets a column vector of real parameters as argument.

  x0: real column vector of initial parameters.

  A, b: Inequality constraints of the parameters `p` with `A * p - b <= 0`.

  Aeq, beq: Equality constraints of the parameters `p` with `A * p - b = 0`.

  lb, ub: Bounds of the parameters `p` with `lb <= p <= ub`.

  nonlcon: Nonlinear constraints. Function returning the current values of nonlinear inequality constraints (constrained to `<= 0`) in the first output and the current values of nonlinear equality constraints in the second output.

  options: structure whose fields stand for optional settings referred to below. The fields can be set by `optimset()`.

  An argument can be set to `[]` to indicate that its value is not set.

  `fmincon` may also be called with a single structure argument with the fields `objective`, `x0`, `Aineq`, `bineq`, `Aeq`, `beq`, `lb`, `ub`, `nonlcon` and `options`, resembling the separate input arguments above. Additionally, the structure must have the field `solver`, set to `"fmincon"`.

  The returned values are the column vector of final parameters x, the final value of the objective function fval, an integer cvg indicating if and how optimization succeeded or failed, and a structure outp with additional information, curently with possible fields: `iterations`, the number of iterations, `funcCount`, the number of objective function calls (indirect calls by gradient function not counted), `constrviolation`, the maximum of the constraint violations. The backend may define additional fields. cvg is greater than zero for success and less than or equal to zero for failure; its possible values depend on the used backend and currently can be `0` (maximum number of iterations exceeded), `1` (success without further specification of criteria), `2` (parameter change less than specified precision in two consecutive iterations), `3` (improvement in objective function less than specified), `-1` (algorithm aborted by a user function), or `-4` (algorithm got stuck).

  #### Options:

  `Algorithm`

    `interior-point`, `sqp`, and `sqp-legacy` are mapped to optims `lm_feasible` algorithm (the default) to satisfy constraints throughout the optimization. `active-set` is mapped to `octave_sqp`, which may perform better if constraints only need to be satisfied for the result. Other algorithms are available with `nonlin_min`.

  `OutputFcn`

    Similar to the setting `user_interaction` — see `optim_doc()`. Differently, `OutputFcn` returns only one output argument, the stop flag.

  `GradObj`

    If set to `"on"`, objf must return the gradient of the objective function as a second output. The default is `"off"`.

  `GradConstr`

    If set to `"on"`, nonlcon must return the Jacobians of the inequality- and equality-constraints as third and fourth output, respectively.

  `HessianFcn`

    If set to `"objective"`, objf must not only return the gradient as the second, but also the Hessian as the third output.

  `Display, FinDiffRelStep, FinDiffType, TypicalX, MaxIter, TolFun, TolX,`

    See documentation of these options in `optim_doc()`.

  For description of individual backends, type `optim_doc ("scalar optimization")` and choose the backend in the menu.

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