int i2 = convert<int>("not an int", cnv).value_or(-1); // after the call i2==-1

if (i2 == -1) process_failure();

The code above is straightforward and self-explanatory but, strictly speaking, is not entirely correct as -1 might be the result of a conversion failure or the successful conversion of the "-1" string. Still, in reality "spare" values (outside the valid/sensible range) are often available to indicate conversion failures. If so, such straightorward deployment might be adequate. Alternatively, it might be not that uncommon to ignore conversion failures altogether and to simply log the event and to proceed with the supplied fallback value.

Applications outside these mentioned categories still require conversion failure reliably detected and processed accordingly. The boost::lexical_cast's (only) answer is to throw on failure and Boost.Convert supports that behavior as well:

try
{
    int i1 = lexical_cast<int>(str);         // Throws if the conversion fails.
    int i2 = convert<int>(str, cnv).value(); // Throws if the conversion fails.
}
catch (...)
{
    process_failure();
}

However, to cater for a wider range of program-flow variations, Boost.Convert adds the flexibility of

optional<int> r1 = convert<int>(str1, cnv); // Does not throw on conversion failure.
optional<int> r2 = convert<int>(str2, cnv); // Does not throw on conversion failure.
// ...
try // Delayed processing of potential exceptions.
{
    int i1 = r1.value(); // Will throw if conversion failed.
    int i2 = r2.value(); // Will throw if conversion failed.
}
catch (boost::bad_optional_access const&)
{
    // Handle failed conversion.
}

// Exceptions are avoided altogether.
int i1 = r1 ? r1.value() : fallback_value;
int i2 = r2.value_or(fallback_value);
int i3 = convert<int>(str3, cnv).value_or(fallback_value);
int i4 = convert<int>(str3, cnv).value_or_eval(fallback_function);

Here boost::optional steps forward as the actual type returned by boost::convert() which until now we avoided by immediately calling its value-accessor methods:

int i1 = boost::convert<int>(str1, cnv).value();
int i2 = boost::convert<int>(str2, cnv).value_or(fallback_value);
int i3 = boost::convert<int>(str3, cnv).value_or_eval(fallback_function);

	Note
	One notable advantage of value_or_eval() over value_or() is that the actual calculation of the fallback_value is potentially delayed and conditional on the success or failure of the conversion.

From the user perspective, boost::lexical_cast processes failure in a somewhat one-dimensional non-negotiable manner. boost::convert takes a more flexible approach. It provides choice and leaves the decision to the user. It is not unimaginable that, on the library level, propagating the conversion-failure exception might be the only available option. On the application level though, in my personal experience, the choice has overwhelmingly been to handle conversion failures locally, i.e. avoiding conversion-failure exception propagation or, better still, avoiding exceptions altogether with program flows similar to:

boost::optional<int> res = boost::convert<int>(str, cnv);

if (!res) log("str conversion failed!");

int i1 = res.value_or(fallback_value);

// ...proceed

and

struct fallback_func
{
    int operator()() const { log("Failed to convert"); return 42; }
};

// Fallback function is called when failed
int i2 = convert<int>(str, cnv).value_or_eval(fallback_func());
int i3 = convert<int>(str, cnv, fallback_func()); // Same as above. Alternative API.