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## Quadratic Minimization with Dense, Structured Hessian

### Take advantage of a structured Hessian

The `quadprog` trust-region-reflective method can solve large problems where the Hessian is dense but structured. For these problems, `quadprog` does not compute H*Y with the Hessian H directly, as it does for trust-region-reflective problems with sparse H, because forming H would be memory-intensive. Instead, you must provide `quadprog` with a function that, given a matrix Y and information about H, computes W = H*Y.

In this example, the Hessian matrix `H` has the structure `H = B + A*A'` where `B` is a sparse 512-by-512 symmetric matrix, and `A` is a 512-by-10 sparse matrix composed of a number of dense columns. To avoid excessive memory usage that could happen by working with `H` directly because `H` is dense, the example provides a Hessian multiply function, `qpbox4mult`. This function, when passed a matrix `Y`, uses sparse matrices `A` and `B` to compute the Hessian matrix product `W = H*Y = (B + A*A')*Y`.

In the first part of this example, the matrices `A` and `B` need to be provided to the Hessian multiply function `qpbox4mult`. You can pass one matrix as the first argument to `quadprog`, which is passed to the Hessian multiply function. You can use a nested function to provide the value of the second matrix.

The second part of the example shows how to tighten the `TolPCG` tolerance to compensate for an approximate preconditioner instead of an exact `H` matrix.

### Step 1: Decide what part of H to pass to quadprog as the first argument.

Either `A` or `B` can be passed as the first argument to `quadprog`. The example chooses to pass `B` as the first argument because this results in a better preconditioner (see Preconditioning).

`quadprog(B,f,[],[],[],[],l,u,xstart,options)`

### Step 2: Write a function to compute Hessian-matrix products for H.

Now, define a function `runqpbox4` that

• Contains a nested function `qpbox4mult` that uses `A` and `B` to compute the Hessian matrix product `W`, where `W = H*Y = (B + A*A')*Y`. The nested function must have the form

`W = qpbox4mult(Hinfo,Y,...)`

The first two arguments `Hinfo` and `Y` are required.

• Loads the problem parameters from `qpbox4.mat`.

• Uses `optimoptions` to set the `HessianMultiplyFcn` option to a function handle that points to `qpbox4mult`.

• Calls `quadprog` with `B` as the first argument.

The first argument to the nested function `qpbox4mult` must be the same as the first argument passed to `quadprog`, which in this case is the matrix B.

The second argument to `qpbox4mult` is the matrix `Y` (of `W = H*Y`). Because `quadprog` expects `Y` to be used to form the Hessian matrix product, `Y` is always a matrix with `n` rows, where `n` is the number of dimensions in the problem. The number of columns in `Y` can vary. The function `qpbox4mult` is nested so that the value of the matrix `A` comes from the outer function. Optimization Toolbox™ software includes the `runqpbox4.m` file.

```function [fval, exitflag, output, x] = runqpbox4 %RUNQPBOX4 demonstrates 'HessianMultiplyFcn' option for QUADPROG with bounds. problem = load('qpbox4'); % Get xstart, u, l, B, A, f xstart = problem.xstart; u = problem.u; l = problem.l; B = problem.B; A = problem.A; f = problem.f; mtxmpy = @qpbox4mult; % function handle to qpbox4mult nested function % Choose algorithm and the HessianMultiplyFcn option options = optimoptions(@quadprog,'Algorithm','trust-region-reflective','HessianMultiplyFcn',mtxmpy); % Pass B to qpbox4mult via the H argument. Also, B will be used in % computing a preconditioner for PCG. [x, fval, exitflag, output] = quadprog(B,f,[],[],[],[],l,u,xstart,options); function W = qpbox4mult(B,Y) %QPBOX4MULT Hessian matrix product with dense structured Hessian. % W = qpbox4mult(B,Y) computes W = (B + A*A')*Y where % INPUT: % B - sparse square matrix (512 by 512) % Y - vector (or matrix) to be multiplied by B + A'*A. % VARIABLES from outer function runqpbox4: % A - sparse matrix with 512 rows and 10 columns. % % OUTPUT: % W - The product (B + A*A')*Y. % % Order multiplies to avoid forming A*A', % which is large and dense W = B*Y + A*(A'*Y); end end```

### Step 3: Call a quadratic minimization routine with a starting point.

To call the quadratic minimizing routine contained in `runqpbox4`, enter

`[fval,exitflag,output] = runqpbox4;`

to run the preceding code. Then display the values for `fval`, `exitflag`, `output.iterations`, and `output.cgiterations`.

```fval,exitflag,output.iterations, output.cgiterations fval = -1.0538e+03 exitflag = 3 ans = 18 ans = 30```

After 18 iterations with a total of 30 PCG iterations, the function value is reduced to

```fval fval = -1.0538e+003```

and the first-order optimality is

```output.firstorderopt ans = 0.0043```

### Preconditioning

Sometimes `quadprog` cannot use `H` to compute a preconditioner because `H` only exists implicitly. Instead, `quadprog` uses `B`, the argument passed in instead of `H`, to compute a preconditioner. `B` is a good choice because it is the same size as `H` and approximates `H` to some degree. If `B` were not the same size as `H`, `quadprog` would compute a preconditioner based on some diagonal scaling matrices determined from the algorithm. Typically, this would not perform as well.

Because the preconditioner is more approximate than when `H` is available explicitly, adjusting the `TolPCG` parameter to a somewhat smaller value might be required. This example is the same as the previous one, but reduces `TolPCG` from the default 0.1 to 0.01.

```function [fval, exitflag, output, x] = runqpbox4prec %RUNQPBOX4PREC demonstrates 'HessianMultiplyFcn' option for QUADPROG with bounds. problem = load('qpbox4'); % Get xstart, u, l, B, A, f xstart = problem.xstart; u = problem.u; l = problem.l; B = problem.B; A = problem.A; f = problem.f; mtxmpy = @qpbox4mult; % function handle to qpbox4mult nested function % Choose algorithm, the HessianMultiplyFcn option, and override the TolPCG option options = optimoptions(@quadprog,'Algorithm','trust-region-reflective',... 'HessianMultiplyFcn',mtxmpy,'TolPCG',0.01); % Pass B to qpbox4mult via the H argument. Also, B will be used in % computing a preconditioner for PCG. % A is passed as an additional argument after 'options' [x, fval, exitflag, output] = quadprog(B,f,[],[],[],[],l,u,xstart,options); function W = qpbox4mult(B,Y) %QPBOX4MULT Hessian matrix product with dense structured Hessian. % W = qpbox4mult(B,Y) computes W = (B + A*A')*Y where % INPUT: % B - sparse square matrix (512 by 512) % Y - vector (or matrix) to be multiplied by B + A'*A. % VARIABLES from outer function runqpbox4prec: % A - sparse matrix with 512 rows and 10 columns. % % OUTPUT: % W - The product (B + A*A')*Y. % % Order multiplies to avoid forming A*A', % which is large and dense W = B*Y + A*(A'*Y); end end```

Now, enter

`[fval,exitflag,output] = runqpbox4prec; `

to run the preceding code. After 18 iterations and 50 PCG iterations, the function value has the same value to five significant digits

```fval fval = -1.0538e+003```

and the first-order optimality is essentially the same.

```output.firstorderopt ans = 0.0043```

### Note

Decreasing `TolPCG` too much can substantially increase the number of PCG iterations.

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